Download - SOME BEARING PROBLEMS IN TRIBOLOGY
SOME BEARING PROBLEMS IN TRIBOLOGY
A Thesis submitted to Gujarat Technological University
for the Award of
Doctor of Philosophy
in
Science-Maths
by
Mohmmadraiyan Mohmmadnur Munshi
Enrolment No.: 149997673011
under supervision of
Dr. Ashok R. Patel
GUJARAT TECHNOLOGICAL UNIVERSITY
AHMEDABAD
NOVEMBER, 2020
SOME BEARING PROBLEMS IN TRIBOLOGY
A Thesis submitted to Gujarat Technological University
for the Award of
Doctor of Philosophy
in
Science-Maths
by
Mohmmadraiyan Mohmmadnur Munshi
Enrolment No.: 149997673011
under supervision of
Dr. Ashok R. Patel
GUJARAT TECHNOLOGICAL UNIVERSITY
AHMEDABAD
NOVEMBER, 2020
ii
© Mohmmadraiyan Mohmmadnur Munshi
iii
DECLARATION
I declare that the thesis entitled “Some bearing problems in Tribology” submitted by me for
the degree of Doctor of Philosophy is the record of research work carried out by me during the
period from March 2015 to March 2020 under the supervision of Dr. Ashok R. Patel,
Associate Professor and Head, General Department, Vishwakarma Government Engineering
College, Ahmedabad, Gujarat and this has not formed the basis for the award of any degree,
diploma, associateship, fellowship, titles in this or any other University or other institution of
higher learning.
I further declare that the material obtained from other sources has been duly acknowledged in
the thesis. I shall be solely responsible for any plagiarism or other irregularities, if noticed in
the thesis.
Signature of the Research Scholar: Date: 26/11/2020
Name of Research Scholar: Mohmmadraiyan M. Munshi
Place: Kalol
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CERTIFICATE
I certify that the work incorporated in the thesis “Some bearing problems in Tribology”
submitted by Shri Mohmmadraiyan Mohmmadnur Munshi was carried out by the candidate
under my supervision/guidance. To the best of my knowledge: (i) the candidate has not
submitted the same research work to any other institution for any degree/diploma,
Associateship, Fellowship or other similar titles (ii) the thesis submitted is a record of original
research work done by the Research Scholar during the period of study under my supervision,
and (iii) the thesis represents independent research work on the part of the Research Scholar.
Signature of Supervisor: Date: 26/11/2020
Name of Supervisor: Dr. Ashok R. Patel
Place: Ahmedabad
v
Course-work Completion Certificate
This is to certify that Mr. Mohmmadraiyan Mohmmadnur Munshi, Enrolment no.
149997673011 is a PhD scholar enrolled for PhD program in the branch Science-Maths of
Gujarat Technological University, Ahmedabad.
(Please tick the relevant option(s))
He has been exempted from the course-work (successfully completed during
M.Phil Course)
He has been exempted from Research Methodology Course only (successfully
completed during M.Phil Course)
He has successfully completed the PhD course work for the partial requirement
for the award of PhD Degree. His performance in the course work is as follows-
Grade Obtained in Research Methodology
(PH001)
Grade Obtained in Self Study Course (Core Subject)
(PH002)
-- AB
Supervisor’s Sign:
Name of Supervisor: Dr. Ashok R. Patel
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Mohmmadraiyan Mohmmadnur Munshi has been examined by us. We undertake the
following:
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under consideration to be published elsewhere. No sentence, equation, diagram, table,
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Signature of the Research Scholar: Date: 26/11/2020
Name of Research Scholar: Mohmmadraiyan M. Munshi
Place: Kalol
Signature of Supervisor: Date: 26/11/2020
Name of Supervisor: Dr. Ashok R. Patel
Place: Ahmedabad
vii
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Signature of the Research Scholar: Date: 26/11/2020
Name of Research Scholar: Mohmmadraiyan M. Munshi
Place: Kalol
Signature of Supervisor: Date: 26/11/2020
Name of Supervisor: Dr. Ashok R. Patel
Place: Ahmedabad
Seal:
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Abstract
All the research in the field of roughness bearing system aims to enhance the designing quality
of products and process. The measurement specification of a system’s roughness and their
analysis helps to understand the concept of roughness and related aspects better. A lot of
phenomena are possible, impacted by roughness of bearing systems which can be better
understood through such research.
Surface roughness plays vital role in the field of Tribology. To measure a random distribution
of the height of the surface is known as its roughness. The roughness of various surfaces at the
concentric position of their slope can be sampled and averaged to find a mean absolute slope.
A relative number of the micro contact areas can be found through this calculation. The
surface roughness is expressed in terms of a stochastically random variable that has a mean,
skewness and variance as non-zero.
Liquids like mercury or hydrocarbon, that are also known as carried liquids, have suspended
magnetic metal particles, rather nano-particles, which are stable and colloidal in nature. A
constant magnetic field can be used to give a stabilized position to the magnetic fluid. This
makes a magnetic fluid a good lubricant. Due to such properties, the expansion of magnetic
fluid leads to utilized in sealing computer hard disks or drives, shaft and rods rotations,
rotating x-ray tubes etc. These fluids also serve as highly efficient as heat controllers in
various systems like electric motors and even hi-fi speaker systems. One of the liquids that
display strong magnetization when exposed to a magnetic field is Ferrofluid. A Ferrofluid can
be developed using three materials; magnetic particles that have a colloidal size, a liquid that
can act as a carrier and a surfactant. A lot of devices that have a magnetic fluid design
including pressure transducers, accelerometers, sensors and others make use of Ferrofluids.
Actuating machines such as energy converters and even electromechanical converters use
Ferrofluids.
Not only engineering, but the applications of magnetic fluids are also relevant and popular in
biomedicine. Some studies have shown significant results by using magnetic fluids for cancer
treatment. The concept here is to soak the tumor in a magnetic fluid with the help of a
changing magnetic field and then heating the tumor.
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Fluid dynamics has the no-slip condition in viscous fluids suggests that by maintaining a solid
boundary, a state of zero relative velocity between the fluid and the boundary can be achieved.
However, some scientists reported cases where this was not always the case. It was general
phenomenon which called slip velocity, where the fluid posses some velocity with respect to
solid boundary; this velocity is identified as a slip velocity. When the difference in the mean
velocities of two separate fluids that are in a pipe, flowing together, is calculated, the slip
velocity of the two fluids can be found. The key characteristic that changes the slip velocity of
a fluid is its density as compared to the other fluid. When the flow is ascending vertical in
nature, the fluid with the lower density moves with higher speed than another one.
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Acknowledgement
I am very much thankful to Almighty for giving me an opportunity to undertake the research
work and enabling me to its completion.
Apart from the grace of almighty, numbers of persons have been extremely generous and
helpful to me during the course of this study. I cannot afford to miss at least few of them who
have been constant source of inspiration and moral support towards the completion of my
thesis.
First and foremost, I am thankful to my beloved, honorable and enthusiastic research
supervisor Dr. Ashok R. Patel (Associate Professor and Head, Vishwakarma Government
Engineering College, Ahmedabad) for his constant motivation which proved to be a real
motivation and encouragement for me to complete this research work. I consider myself very
fortunate that, I got an opportunity to work under his guidance. I express my deep sense of
gratitude to Dr. Gunamani B. Deheri (Former Associate Professor, Sardar Patel University,
Vallabh Vidyanagar) for his valuable, untiring guidance and constructive criticism which have
enabled me to successfully complete this study and teach a lot many things. My thankfulness
to him goes beyond this formal acknowledgement and cannot be fully expressed in my words.
He always cleared my doubts in the research and provides vital suggestions. My interaction
with him in these years has given me an insight into the subject. The experience and learning
that I had with him is enormous and will stay with me throughout my life.
I thank my Doctoral Progress committee members, Dr. Himanshu C. Patel (Registrar, Indian
Institute of Teacher Education, Gandhinagar) and Dr. Mukesh E. Shimpi (Associate Professor,
Birla Vishvakarma Mahavidyalaya Engineering College, Vallabh Vidyanagar) who involves
with me indirectly with their expertise and enrich my work with their fruitful comments. They
always had an apt the question, the answer for which would always become the next step of
my research.
The road of my PhD started with the unconditional love of Dr. Jayesh K. Ratnadhariya
(Principal, Hasmukh Goswami College of Engineering, Ahmedabad) who always pour the
positive vibes in the field of my academic career.
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It is my failure if I do not acknowledge Dr. Ravikumar K. (Director, Shankersinh Vaghela
Bapu Institute of Technology, Gandhinagar) and Prof. G. N. Patel (Former Technical Advisor,
Alpha college of engineering and Technology) whose constant support and grace as well
wishers made me to reach at this platform.
It gives me immense pleasure to convey warmer regards to our trustees of Alpha College of
Engineering and Technology, Smt. Sangita Raje, a personality with accuracy who works like a
clock to achieve the goals not for her but for others, Shri Laxmanbhai Patel, pillar of Alpha
Education Foundation, and Shri Sureshbhai Patel an active member of Alpha Education
Foundation for their active involvement towards faculty development. Also, I would like to
give my special thanks to Dr. Santosh S. Kolte (Principal, Alpha College of Engineering and
Technology, Khatraj, Kalol) for his valuable cooperation and support.
I have been always blessed with magnificent senior research scholars who always walk with
me side by side and provide a stimulation as well as fun filled environment. Dr. Paresh A.
Patel (Madhav Science School, Ahmedabad) who creates a base, Dr. Nitin D. Patel (Assistant
Professor, Anand Agricultural university, Anand) always there for me 24/7, Dr. Jimit R. Patel
(Assistant Professor, Charotar University of Science and Technology) a source of resolve the
problems whenever I struck and Dr. Nimeshchandra S. Patel (Assistant Professor, Dharmsinh
Desai University, Nadiad) who plays a vital role behind the curtain always.
My Special appreciations to Dr. Yogini D. Vashi (Assistant Professor, Applied Science and
Humanities Department, Alpha College of Engineering and Technology), Mr. Maulik Barot
(Assistant Professor, Applied Science and Humanities Department, Alpha College of
Engineering and Technology) and Mr. Bhavesh A. Patel (Assistant Professor, Mechanical
Engineering Department, Alpha College of Engineering and Technology). It was their
unconditional love and effort which instilled the positivity with their respected support and
suggestions.
I would like to express my regards to the Vice Chancellor Dr. Navin Sheth, Dean of PhD
programme, Registrar and entire staff members of PhD section from Gujarat Technological
University, who directly or indirectly support me in this journey because any seed needs a
land to grow.
I also express my sincere gratitude to Ms. Hiteshree Dudani for her constant help to shape my
work in a proper way; it was very difficult to enrich my work without her support. At this
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moment how I can forget my dear and near friend Mr. Chirag Shah (SEO & Digital Marketing
Consultant, Ahmedabad) who always stand for me in any moment to show the track in a fussy
time.
Finally, my deep and sincere gratitude to my family for their continuous and unparalleled
love, help and support. I am forever indebted to my parents for giving me the opportunities
and experiences that have made me who I am. They selflessly encouraged me to explore new
directions in life and seek my own destiny. This journey would not have been possible if my
papa Mr. Mohmmadnur F. Munshi didn’t walks with me in pros and cons of my life, and I
dedicate this milestone to him. At this moment if I do not quote the name of my two pillar
who inspired me and taught me to remain in race at any phase of life Mr. Mohmmadumar F.
Munshi and Mr. Abdulgaffar F. Munshi. If these two people would not be there then I cannot
be I what I am. I know shadow does not need any identity but without its reflection a person
can’t get the idea of his or her existence and so it’s not a formal part but without the support of
my beloved wife Gufrana, It would difficult to finish this task because when I was in block of
research she support me with her positive belief towards my dream.
My special regards to my teachers because of whose teaching at different stages of education
has made it possible for me to see this day. Because of their kindness I feel, was able to reach
a stage where I could write this thesis.
Lastly, I thank all my friends, my colleagues and my well-wishers who have directly or
indirectly contributed to this research work. This success belongs not only from me but to all
those who supported me with their loving, care and patience.
Last but not least, I would like to address special thanks to the reviewers of my thesis, for
accepting to read and review this thesis and giving approval of it. I would like to appreciate all
the researchers whose works I have used, initially in understanding my field of research and
later for updates.
Mohmmadraiyan M. Munshi
xvi
Dedicated to my beloved
PAPA and my respected
TEACHERS
xvii
Contents
Abstract xi
Acknowledgement xiii
List of Symbols xx
List of Figures xxiv
List of Tables xxvi
1 Introduction 1
1.1 Abstract ………………………………………………………………………………....... 1
1.2 Brief description on the state of the art of the research topic ……………………………. 3
1.3 Definition of the problem ………………………………………………………………... 5
1.4 Objective and scope of work …………………………………………………………….. 5
1.4.1 Research objectives ……………………………………………………………… 5
1.4.2 Scope of the study …………………………………………………………........... 6
1.5 Original contribution by the thesis ………………………………………………………. 7
1.6 Methodology of research, results/comparisons …………………………………….......... 7
1.7 Achievements with respect to objectives ………………………………………………… 9
2 Bearing theory and governing equations 10
2.1 Equation of state …………………………………………………………………………. 10
2.2 Darcy’s law …………………………………………………………………………......... 10
2.3 Equation of motion …………………………………………………………………......... 11
2.4 Continuity equation …………………………………………………………………........ 12
2.5 Reynolds equation for two-dimensional flow ………………………………………......... 14
2.5.1 Using the Navier-Stokes and continuity equations ………………………............. 14
2.5.2 Equation for short bearing ……………………………………….......................... 19
2.5.3 Equation for infinitely long parallel plates ………………………………………. 20
2.5.4 Equation for plane slider bearing ……………………………................................ 20
2.5.5 Equation for parallel circular plate ………………………………………………. 20
2.5.6 Equation for rectangular plate on a plane surface ……………….......................... 21
2.5.7 Equation for infinitely long rectangular plate ……………………………………. 21
2.5.8 Equation for complete cone ……………………………………………………… 21
2.5.9 Equation for truncated cone ……………………………………………………… 21
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2.5.10 Equation for parallel-step-pad slider bearing ……………………………………. 22
2.5.11 Equation for circular step bearing ………………………………........................... 22
2.5.12 Equation for circular disks ……………………………………………………….. 22
2.5.13 Equation for Neuringer-Rosensweig model ……………………………………... 22
2.5.14 Equation for Shliomis model …………………………………….......................... 24
2.5.15 Equation for Jenkins model ……………………………………………………… 25
3 Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with
slip velocity
27
3.1 Introduction ………………………………………………………………………………. 27
3.2 Analysis ………………………………………………………………………………… 29
3.3 Results and discussion …………………………………………………………………… 32
3.4 Validation ……………………………………………………………………………… 39
3.5 Conclusions ………………………………………………………………………………. 40
4 Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider
bearing
42
4.1 Introduction ………………………………………………………………………………. 42
4.2 Analysis ………………………………………………………………………………….. 44
4.3 Results and discussion …………………………………………………………………… 47
4.4 Conclusions ……………………………………………………………………………… 54
5 Numerical modelling of Shliomis model based ferrofluid lubrication performance in
rough short bearing
55
5.1 Introduction ………………………………………………………………………………. 55
5.2 Analysis ………………………………………………………………………………….. 56
5.3 Results and discussion ………………………………………………………………….... 62
5.4 Validation ……………………………………………………………………………… 70
5.5 Conclusions ………………………………………………………………………………. 72
6 Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
73
6.1 Introduction ………………………………………………………………………………. 73
6.2 Analysis ………………………………………………………………………………… 75
6.3 Results and discussion …………………………………………………………………… 81
6.4 Conclusions ………………………………………………………………………………. 89
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7 A study of ferrofluid lubrication based rough sine film slider bearing with assorted
porous structure
90
7.1 Introduction ………………………………………………………………………………. 90
7.2 Analysis ………………………………………………………………………………….. 92
7.3 Results and discussion …………………………………………………………………… 98
7.4 Validation ………………………………………………………………………………... 105
7.5 Conclusions ………………………………………………………………………………. 106
8 General conclusion and future scope 108
References 111
List of Publications 123
xx
List of Symbols
a Inlet-outlet ratio in the case of slider bearing, dimension of the bearing in the case
of truncated conical plates and outer radius in the case of circular plates
A material constant parameter
b dimension of the bearing in the case of truncated conical plates and inside radius
in the case of circular plates
B breadth in the case of slider bearing
B magnetic induction vector
c maximum deviation from mean level
E expectancy operator
E electric field intensity vector
1 2, ,g g g function of different parameters
h film thickness (mm)
mh mean film thickness (mm)
sh deviation from mean level
0h
central film thickness(mm)
1h maximum film thickness (mm)
2h minimum film thickness (mm)
h non dimensional film thickness
0h
squeeze velocity (m/s)
H external magnetic field vector (Gauss)
H magnitude of H (N/A.m)
0H constant magnetic field
H
porous layer thickness
I sum of moments of inertia of the particle per unit volume (N s2 m
−2)
xxi
J electric current density vector
Bk Boltzmann constant (J K−1
)
K permeability of porous facing in the case of slider bearing and aspect ratio b/a
(width/height) in the case of truncated conical plates
L length of the bearing
m aspect ratio 1 2 2/h h h
M magnetization vector
0M equilibrium magnetization (A m−1
)
n number of particles per unit (m−3
)
p pressure in the film region (N m−2
)
p
pressure in the porous region
p non-dimensional film pressure
q thermal factor
q ( , , )u v w is the fluid velocity vector
Q integrating constant
R universal gas constant
r radial coordinate
s slip parameter (m-1
)
s dimensionless slip parameter
S internal angular momentum vector
t time
T Temperature (K)
U uniform velocity in the direction of x-axis (m/s)
, ,u v w the velocities components in ,x y and z directions (m/s)
, ,a a au v w the velocities components of upper surfaces in ,x y and z directions (m/s)
, ,b b bu v w the velocities components of lower surfaces in ,x y and z directions (m/s)
V velocity approach, volume of the particle
0w values of w at 0z
xxii
hw values of w at z h
W load capacity (N)
W non-dimensional load capacity
,x z the bearing width and length coordinates
y fluid film thickness coordinate
variance (mm)
non- dimensional variance
1 squeeze parameter
skewness (mm3)
skewness in dimensionless form
slip coefficient
fluid viscosity
0 viscosity of the main liquid (N s m−2
)
electrical conductivity
coefficient of bulk viscosity
magnetic moment of a particle
0 permeability of the free space (N A−2
)
magnetic susceptibility of particles (mm3/kg)
dimensionless magnetization parameter
Langevin’s parameter (>1)
lubricant density (N.sec2/m
4)
standard deviation (mm)
dimensionless standard deviation
magnetization parameter
B Brownian relaxation time parameter
S magnetic moment relaxation time parameter
inclination angle ( )
xxiii
volume concentration of the particles nV
dimensionless porosity
dimensionless conventional porosity
semi vertical angle of the cone
fluid vorticity
xxiv
List of Figures
1.1 Configuration of the bearing system …………………………………………………………. 8
2.1 Velocities and densities for mass flow balance through a fixed volume element in two
dimensions ……………………………………………………………………………………. 13
2.2 Viscous flow ………………………………………………………………………………….. 15
3.1 Configuration of truncated conical plates …………………………………………………….. 29
3.2 Profile of W with regards to s ………………………………………………………………. 34
3.3 Profile of W with regards to K ……………………………………………………………… 35
3.4 Profile of W with regards to ……………………………………………………………... 36
3.5 Profile of W with regards to ……………………………………………………………... 37
3.6 Profile of W with regards to ……………………………………………………………... 38
3.7 Profile of W with regards to ……………………………………………………………… 38
4.1 Physical geometry of the bearing system …………………………………………………….. 44
4.2 Profile of W with regards to
……………………………………………………………... 50
4.3 Profile of W with regards to ……………………………………………………………... 51
4.4 Profile of W with regards to ……………………………………………………………... 52
4.5 Profile of W with regards to ……………………………………………………………… 53
4.6 Profile of W with regards to ……………………………………………………………... 53
5.1 Configuration of the bearing system …………………………………………………………. 56
5.2 Profile of W with regards to ……………………………………………………………… 65
5.3 Profile of W with regards to ……………………………………………………………… 67
5.4 Profile of W with regards to m ……………………………………………………………... 68
5.5 Profile of W with regards to ……………………………………………………………... 69
5.6 Profile of W with regards to ……………………………………………………………... 70
5.7 Profile of W with regards to ……………………………………………………………… 70
6.1 Configuration of the bearing system …………………………………………………………. 75
6.2 Profile of W with regards to ……………………………………………………………… 83
6.3 Profile of W with regards to q ……………………………………………………………… 85
6.4 Profile of W with regards to m ……………………………………………………………... 86
6.5 Profile of W with regards to ……………………………………………………………... 87
xxv
6.6 Profile of W with regards to ……………………………………………………………... 88
6.7 Profile of W with regards to ……………………………………………………………… 88
7.1 Configuration of a sine film porous slider bearing including squeeze action ………………... 93
7.2 Structure model of porous sheet given by Kozeny‐Carman ………………………………….. 96
7.3 Profile of W with regards to
……………………………………………………………... 100
7.4 Profile of W with regards to ……………………………………………………………... 101
7.5 Profile of W with regards to ……………………………………………………………... 102
7.6 Profile of W with regards to …………………………………………………………….... 102
7.7 Profile of W with regards to K ……………………………………………………………... 103
7.8 Profile of W with regards to ……………………………………………………………... 103
7.9 Profile of W with regards to l …………………………………………………………….... 104
7.10 Profile of W with regards to , and for the comparison of and ……………... 105
xxvi
List of Tables
3.1 Comparison of W calculated for ……………………………………………………………. 39
3.2 Comparison of W calculated for ……………………………………………………………. 39
3.3 Comparison of W calculated for ……………………………………………………………. 39
3.4 Comparison of W calculated for ……………………………………………………………. 40
3.5 Comparison of W calculated for ……………………………………………………………. 40
3.6 Comparison of W calculated for K ……………………………………………………………. 40
5.1 Comparison of W calculated for ……………………………………………………………. 70
5.2 Comparison of W calculated for ……………………………………………………………. 71
5.3 Comparison of W calculated for ……………………………………………………………. 71
5.4 Comparison of W calculated for m ……………………………………………………………. 71
5.5 Comparison of W calculated for B ……………………………………………………………. 71
7.1 Comparison of W calculated for ……………………………………………………………. 105
7.2 Comparison of W calculated for ……………………………………………………………. 106
7.3 Comparison of W calculated for ……………………………………………………………. 106
7.4 Comparison of W calculated for ……………………………………………………………. 106
7.5 Comparison of W calculated for ……………………………………………………………. 106
1
CHAPTER 1
Introduction
1.1 Abstract
All the research in the field of roughness bearing system aims to enhance the designing quality
of products and process. The measurement specification of a system’s roughness and its
analysis helps to understand the concept of roughness and related aspects better. A lot of
phenomena are possible, impacted by roughness of bearing systems which can be better
understood through such research.
Surface roughness plays vital role in the field of Tribology. To measure a random distribution
of the height of the surface is known as its roughness. The roughness of various surfaces at the
concentric position of their slope can be sampled and averaged to find a mean absolute slope.
A relative number of the micro contact areas can be found through this calculation. The
surface roughness is expressed in terms of a stochastically random variable that has a mean,
skewness and variance as non-zero.
Liquids like mercury or hydrocarbon, that are also known as carried liquids, have suspended
magnetic metal particles, rather nano-particles, which are stable and colloidal in nature. A
constant magnetic field can be used to give a stabilized position to the magnetic fluid. This
makes a magnetic fluid a good lubricant. Due to such properties, the expansion of magnetic
fluid leads to utilized in sealing computer hard disks or drives, shaft and rods rotations,
rotating x-ray tubes etc. These fluids also serve as highly efficient as heat controllers in
various systems like electric motors and even hi-fi speaker systems. One of the liquids that
Introduction
2
display strong magnetization when exposed to a magnetic field is Ferrofluid. A Ferrofluid can
be developed using three materials; magnetic particles that have a colloidal size, a liquid that
can act as a carrier and a surfactant. Lot of devices that have a magnetic fluid design including
pressure transducers, accelerometers, sensors and others make use of Ferrofluids. Actuating
machines such as energy converters and even electromechanical converters use Ferrofluids.
Not only engineering, but the applications of magnetic fluids are also relevant and popular in
biomedicine. Some studies have shown significant results by using magnetic fluids for cancer
treatment. The concept here is to soak the tumor in a magnetic fluid with the help of a
changing magnetic field and then heating the tumor.
Fluid dynamics has the no-slip condition in viscous fluids suggests that by maintaining a solid
boundary, a state of zero relative velocity between the fluid and the boundary can be achieved.
However, some scientists reported cases where this was not always the case. It was general
phenomenon which called slip velocity, where the fluid possesses some velocity with respect
to solid boundary; this velocity is identified as a slip velocity. When the difference in the mean
velocities of two separate fluids that are in a pipe, flowing together, is calculated, the slip
velocity of the two fluids can be found. The key characteristic that changes the slip velocity of
a fluid is its density as compared to the other fluid. When the flow is ascending vertical in
nature, the fluid with the lower density moves with higher speed than another one.
This study has attempted to scrutinize the bearing performance of a rough bearings assisted by
Ferrofluid with the help of numerical modelling of Shliomis model as well as Neuringer and
Rosensweig. The transverse and longitudinal roughness are calculated stochastically by
averaging the Christensen and Tonder model. A non-zero mean is assumed for the probability
density function for the random variable that determines the roughness of the bearing which is
symmetrical. One of the equations that can aid the calculation of dependent permeability
which is influenced by factors like pore shape, porosity, tortuosity and specific surface is
Kozeny-Carman’s model. The Beavers and Joseph model is used to study the effects caused
by slip velocity. The Tipei model and the Shliomis model have been used to derive a new
structure for the Reynolds’ equation which can be used to calculate thermal variation.
The attempt is made to create a more pragmatic and applicable situation. Expressions that can
signify dimensionless form of pressure and bearing load carrying capacity are found using
Reynolds’ equation. The load carrying capacity equation is then solved numerically with the
Abstract
3
help of Simpson’s 1/3 rule to analyze the impact on the bearing system. From the graphical
study representation, it can be concluded that a Ferrofluid lubrication based on the Shliomis
model can significantly neutralize the negative effects of the bearing’s roughness on its load
carrying capacity.
1.2 Brief Description on the State of The Art of the Research Topic
The discipline that studies the phenomenon occurring when two objects that are relatively
moving and are in contact with each other is known as Tribology. A UK-based committee in
1966 first used the term “Tribology” (Dowson, 1979).
A very noteworthy work in the field, “History of Tribology-the Bridge between Classical
Antiquity and the 21st century” was given by (Bartz, 2001). This work gave an extensive
account of the history of Tribology and the way it has evolved through the years. Tribology,
which is interestingly amongst the earliest phase of engineering sciences, has been compared
to various classical disciplines. When early humans started looking for ways to reduce manual
labor in load carrying, the science of Tribology began. In fact, this science has been relevant
since the time the first wheel was invented.
The research tries to come over with various applications through different bearings which can
be utilized in digitalized world. Various industrial applications including aerospace and
aeronautical industries, nuclear and civil engineering, modern construction engineering
amongst others make use of conical plates as crucial constitutional elements. The dynamic
response of these conical plates is significantly impacted by various fluids (stationary or
flowing) that they work with. That is why, it is crucial to study the behavior generated by
different load types in order to ensure safe functioning in applications. It is also clear that
same as the conical bearing, Slider bearing has its own metallic aspects, a lot of applications in
various fields including clutch plates, automobile transmissions and domestic appliances. On
the other hand porous bearings are also used in horsepower motors of hair dryers, record
players, vacuum cleaners, tape recorders, sewing machines, water pumps, etc. (Patel & Deheri,
2018).
Christensen and Tonder (1969a,b, 1970) used a stochastic concept and came up with a new
model for lubricated surfaces with striated roughness using an averaging film. They derived
Introduction
4
the stochastic Reynolds’ equation and used the results to study the impact of surface
roughness on the load bearing capacity in a rough bearing system. Many famous books of the
field (Bhat, 2003; Hamrock, 1994; Majumdar, 2008) discuss the Reynolds’ equation and try to
derive an exact solution to it by using different basic film geometries. The last decade has seen
a considerable shift wherein many tribological researches have been dedicated to study surface
roughness and its impact of hydrodynamic lubrication. This is because every solid surface
carries some amount of surface roughness, the height of which is usually parallel to the mean
separation between lubricated contacts. As many researchers have suggested, studying surface
roughness will help to improve the performance of bearing system. Due to this reason, many
researchers (Andharia et al., 2001; Naduvinamani et al., 2015; Patel et al., 2012b; Shukla &
Deheri, 2017; Thakkar et al., 2019) studied the performance of various bearing systems using
the stochastic concept of (Christensen & Tonder, 1969a,b, 1970).
All the particles undergo a body force when subjected to a magnetic field, resulting in the drag
to flow. Therefore, for industrial application, the study of different combination of materials
with magnetic fluid is of primary importance (Patel et al., 2017c). Some researchers (Bhat &
Deheri, 1991a; Neuringer & Rosensweig, 1964; Shah & Bhat, 2002; Shimpi & Deheri, 2012a;
Snyder, 1962) have also used magnetic fluid as a lubricant in order to aid the tribological
performance of a sliding interface.
Furthermore, porosity was introduced in an attempt to decrease the friction. Morgan and
Cameron (1957) were the first investigators to study the hydrodynamic lubrication theory of
bearings with porous structure. Darcy’s law is generally used to determine the porosity.
Porous metallic materials have a lot of applications including vibration and sound absorption,
light materials, heat transfer media, sandwich core for different panels, various membranes
and during the last years as suitable biomaterial structures for design of medical implants.
Beavers and Joseph (1967) provide some boundary conditions that were empirical in nature
which gave a coefficient of slip known as . These conditions can be used to calculate the
non-zero type of interfacial velocity when the flow in a porous medium increases
significantly. In the given context, the efficiency of the thermal impact cannot be marginalized
which is why, Tipei (1962) performed an experimental study which suggested that the
viscosity-temperature relationship is substitutable by a establishing a relationship between the
viscosity and the film thickness. The study also suggested that least film thickness is
Brief description on the state of the art of the research topic
5
associated with highest temperature. In the modern age efficacy of thermal effect also
introduced with new shape.
1.3 Definition of the Problem
When there is an increased amount of contact between two metallic surfaces that are non-
lubricated or dry, it causes friction which leads to wear and tear. This not only leads to energy
wastage due to friction but even the material of the system is compromised due to the wear
and tear. Lubricants like viscous fluid or liquid metal or others are used to reduce the friction.
These substances create a space between the two surfaces in which they can function smoothly
with minimal efforts. The type of lubrication to be used is based on different aspects
including, the surface geometry, the load to be carried, relative velocity of both the surfaces
and the characteristics of the lubricant amongst other.
This study aims to explore the impact created by slip velocity, porosity, assorted porous
structure (Carman, 1937) and variation in viscosity along with the roughness longitudinal as
well as transverse of a bearing surface on a Ferrofluid lubrication of different magnetic fluid
flow model i.e. Shliomis model and Neuringer-Rosensweig model for various bearing system.
The average pressure of a slider bearing with a rough surface is calculated using the given
averaged Reynolds’ equation and is explained by (Bhat, 2003).
3 2 3 2
0 0
1 112 12 6 12
2 2
h hh KH p H h KH p H U
x x y y x t
(1.1)
1.4 Objective and Scope of Work
1.4.1 Research Objectives
This study aims to understand the way pressure and system’s load bearing capacity of this
mathematical model function when a Ferro-lubricant is used instead of a conventional or
regular lubricant.
The effect of slip velocity is going to be examined in:
• Ferrofluid squeeze film in longitudinally rough truncated conical plates.
Introduction
6
• Ferrofluid based longitudinally rough porous plane slider bearing.
This study tries to carry out new dimension with the help of Shliomis model and claim for
better result, and which is going to one hand experience in,
• Ferrofluid lubrication performance in rough short bearing.
• Lubrication of rough short bearing by Ferrofluid considering viscosity variation effect.
This research tries to explain with different film geometries to check the potency of bearing
system and its result.
Furthermore, our goal in the concluding chapter is a detailed scrutiny of Ferrofluid lubrication,
most effectively on the basis of rough sine film slider bearing with assorted porous structure.
1.4.2 Scope of the Study
This study aims to perform a comprehensive analysis of the following:
• The impact caused by deformation in the load bearing capacity of different bearing
systems.
• These models of magnetic fluid flow (Jenkins, 1972; Neuringer & Rosensweig, 1964;
Shliomis, 1974) can be compared so as to know in which particular model load bearing
capacity is in high proportion.
• In future, the researcher may focus on applying double layered porous structure to the
various bearings.
• Possibilities for the application of hydromagnetic lubrication to the bearings to
improvise their load carrying capacity have been examined.
• It is also possible to study the theoretical implications concerning the impact of a
system’s roughness on the type and features of lubrication used with the help of
micropolar fluid.
• Ample of scope to front forward with profile of the piston top compression ring face
which is assumed to be a parabola is also found.
• The Jenkins model of fluid flow may be used in order to study the ways in which
deformation can impact different types of bearing systems.
• The impact caused by couple stress is also studied with the help of magnetic fluid
flow.
Objective and scope of work
7
• We have still opportunities are there to explore the research on annular plates with all
the parameters which were utilized in the study.
• Analysis of the surface topology of the bearing system.
By focusing on such a diverse range of topics, this study becomes relevant to various different
streams of engineering and science including physics, material science, mechanical
engineering, mathematics, etc.
1.5 Original Contribution by the Thesis
This thesis modifies and adapts a mathematical model which helps study:
• Influence of Ferrofluid lubrication on longitudinally rough truncated conical plates
with slip velocity.
• Effect of slip velocity on a Ferrofluid based longitudinally rough porous plane slider
bearing.
• Numerical modelling of Shliomis model based Ferrofluid lubrication performance in
rough short bearing.
• Lubrication of rough short bearing on Shliomis model by Ferrofluid considering
viscosity variation effect.
• A study of Ferrofluid lubrication based rough sine film slider bearing with assorted
porous structure.
The graphical method is used to calculate the results. These results are also compared
holistically in order to find the various criteria that would increase the system’s performance.
1.6 Methodology of Research, Results/Comparisons
The following assumptions were considered (Deheri & Patel, 2006)
• The lubricant flow is considered laminar and lubricant film is assumed to be
isoviscous.
• There are no external fields of force acting on the fluid. While magnetic and electric
forces are not present in the flow of non conducting lubricants, forces due to
Introduction
8
gravitational attraction are always present. However, these forces are small compared
to the viscous force involved.
• The flow is considered steady and temperature changes of the lubricant are neglected.
• The bearing surfaces are assumed to be perfectly rigid so that elastic deformations of
the bearing surfaces may be neglected.
• Bearing surfaces are assumed to be perfectly smooth or even when there is surface
roughness it is of very small order of magnitude in comparison with the minimum film
thickness.
• The thickness of the lubricant film is very small when compared to the dimensions of
the bearing.
• The lubricant velocity along the transverse direction to the film is considered small
enough.
• Velocity gradients and indeed the second derivatives along the direction transverse to
the film are predominant as compared to those in the plane of the film.
• The lubricant inertia is considered negligible.
• The porous matrix of the bearing surface is assumed to be homogeneous and isotropic.
• Darcy’s law is assumed to govern the lubricant flow within the porous matrix, while no
slip condition is taken at the porous matrix-film interface.
FIGURE 1.1 Configuration of the bearing system (Patel & Deheri, 2013a)
Thus, it is considered to be 1-D problem. Various parameters of roughness are added at
different stages, like mean, standard deviation and skewness, roughness pattern parameter for
transverse and longitudinal as well, of the rough surface and magnetization parameters. This
Methodology of research, results/comparisons
9
allows the calculation of an average pressure for the system present on the area of contact. We
can derive the system’s load carrying capacity along with the pressure through this
calculation. Existing researches have been used to verify and justify the findings of this study.
Simpson’s one-third rule having step size 0.2 is used to work on the calculations of the
integrals. The findings of the study along with the relations found between parameters are
plotted on a graph and are also represented tabular.
1.7 Achievements with respect to Objectives
The model of (1.1) by (Bhat, 2003) has been adapted to achieve the aim:
The adapted model has been solved while maintaining appropriate boundary conditions
including parameters of roughness (e.g. mean, standard deviation, skewness), roughness
pattern (e.g. longitudinal or transverse), lubricant type (e.g. magnetic lubricant or conventional
lubricant), magnetic parameter, shape of bearing geometry etc. The study and analysis of
different models revealed some noteworthy findings which are:
• The longitudinally surface roughness can be more adoptable as compared to transverse
surface roughness when no slip is involved.
• Magnetic strength in appropriate measures can be used to nullify the impact of the
thermal effect.
• When we used magnetic fields, Ferrofluid increase the capacity of various bearings in
contrast to the systems functioning with conventional bearing.
• At the time when a sine film profile is used to design the slider bearing, it enhances the
bearing capacity than in the case of inclined slider bearing.
• On the contrary thing to be understood is that a constant magnetic field shows a
positive effect on the bearing capacity in the Shliomis model while the same is not true
for Neuringer-Rosensweig Ferrofluid flow model.
10
CHAPTER 2
Bearing Theory and Governing Equations
2.1 Equation of State
The specific details of the fluid’s state are an essential requirement for a phenomenological
consideration. This can be found with the help of the equation of state. In case of an
incompressible fluid, it is:
constant (2.1)
On the other hand, the Boyle-Mariotte law as given below is used in the case of a perfect gas
with isothermal pressure variations:
p RT (2.2)
Where R denotes the universal gas constant.
In the case of compressible lubricant, an assumption of being proportional to p is made.
However, when the lubrication is liquid, it is difficult to ascertain one specific equation of
state.
2.2 Darcy’s Law
In 1856, Darcy first introduced the equation which governs the motion of a fluid in porous
vertical column. It is given as:
Bearing theory and governing equations
11
Ku p
(2.3)
Here u denotes the space averaged velocity, also known as the Darcian velocity, K denotes
the porous region’s permeability, denotes the viscosity coefficient and p is the porous
region’s pressure.
2.3 Equation of Motion
The law of momentum conservation when used in the context of a fluid placed in a control
volume suggests that the forces that are applied to the fluid are equal to the outflow rate of
momentum. The mathematical equation which explains this scenario in the case of laminar,
continuum, isoviscous, Newtonian and compressible fluid flow in which case,
electromagnetic, gravitational or other body forces are considered to be negligible is:
2. .pt
qq q q q (2.4)
here is known as the viscosity coefficient of the given fluid while is known as the bulk
viscosity coefficient. It is usually understood that these two are related by:
3 2 0 (2.5)
Along with Navier’s first derived (2.4) which was in 1821, Stokes also came up with same
equation independently in 1845. Thus, they are called Navier-Stokes equations. The first term
written on the left side of the equation denotes the temporal acceleration while the second one
represents convective inertia. The first term given on the right side is a result of pressure while
the others are resultant of viscous forces. However, in case the fluid is of the incompressible
type, like in the case of majority of liquid lubricants, then:
. 0 q (2.6)
While (2.4) is simplified as
2. pt
qq q q (2.7)
Continuity equation
12
When a lubricant which is electrically conducting is used to apply a large electromagnetic
field of the external type, the circulating currents which are induced, increase. They, in turn,
interact with the magnetic field which creates Lorentz force, a distinct body force. This
additional electromagnetic-type pressurization propels the fluids placed between the bearing
surfaces. In this case, the modified Navier-Stokes equation becomes:
2. pt
qq q q J B (2.8)
Here J is density of the electric current while B is the vector of magnetic induction. Here,
Ohm’s law and Maxwell’s equations are considered. These are,
0 B J (2.9)
. 0 B (2.10)
J E q B (2.11)
0 E (2.12)
. 0 E (2.13)
Here E is the vector of intensity of the electric field, denotes electrical conductivity while
0 represents the lubricants’ magnetic permeability.
2.4 Continuity Equation
The Navier-Stokes equations comprise of three equations and four unknowns, represented by
, ,u v w and p . The viscosity and density of a fluid in this case can be represented as pressure
and temperature functions. The continuity equation provides a fourth distinct equation. The
mass conservation principle suggests that the total mass outflow from any given fluid volume
should be equal to the mass reduction in the volume. This can be calculated by Fig. 2.1. The
mass flow per unit time and area through any given surface can be derived by finding the
product of velocity normal to the surface and density. Hence x component of mass flux given
per unit area placed at volume’s center is u . However, the given flux changes from one
Bearing theory and governing equations
13
point to another as mentioned in Fig. 2.1. Thus, the net outflow of mass per unit time can be
calculated by:
FIGURE 2.1 Velocities and densities for mass flow balance through a fixed volume element in two dimensions
(Majumdar, 2008)
1 ( ) 1 ( )...
2 2
1 ( ) 1 ( )...
2 2
u wu dx dz w dz dx
x z
u wu dx dz w dz dx
x z
(2.14)
and this should be equal to the rate of mass decrease within the element
dx dzt
(2.15)
When simplified, this becomes
0u wt x z
(2.16)
When direction y is also included, the resultant continuity equation becomes:
Continuity equation
14
0u v wt x y z
(2.17)
If the force density is considered to be a constant, the continuity equation changes to:
0u v w
x y z
(2.18)
2.5 Reynolds Equation for Two-Dimensional Flow
2.5.1 Using the Navier-Stokes and continuity equations
The generalized version of the Reynolds’ equation is another pressure equation used mostly in
the hydrodynamic lubrication theory. The Navier-Stokes equation and the continuity equations
can be used to deduce the generalized Reynolds’ equation with certain assumptions. The
Reynolds’ equation uses a lubricant’s density, viscosity and film thickness as its parameters.
The Navier-Stokes equation can be represented as:
22
3
22
3
Du p u u v w u v w uX
dt x x x x y z y y x z x z
Dv p v u v w v wY
dt y y y x y z z z y
22
3
u v
x y x
Dw p w u v w w u y wZ
dt z z z x y z x x z y z y
(2.19)
The terms on the left side of the given equation signify terms of inertia while the right-side
ones are the pressure gradients, body forces and viscous terms.
Since (2.19) has four unknown terms , ,u v w and p , a different equation is crucial for deriving
these unknowns. The fourth equation in this case is the continuity equation. It can be
represented in the Cartesian co-ordinates as:
0u v wt x y z
(2.20)
Bearing theory and governing equations
15
The equation derived from this is applicable to incompressible as well as compressible
lubricants.
The following assumptions are to be made in this case:
• Body force terms and inertia are negligible in comparison to pressure and viscous
terms.
• The pressure variation across the fluid film is zero, which means 0p y .
• The fluid-solid boundaries have no slip (see Fig. 2.2).
• There are no external forces acting on the film.
• The flow is of viscous and laminar nature (see Fig. 2.2).
FIGURE 2.2 Viscous flow (Majumdar, 2008)
Because of the fluid film’s geometry the derivatives of u and w with respect to y are
significantly larger than other velocity components’ derivatives.
The film’s height which is denoted by h is significantly smaller than the bearing length l .
Using these assumptions, (2.19) can take the form:
p u
x y y
p w
z y y
(2.21)
Reynolds equation for two-dimensional flow
16
Since p is a function of x and z , (2.21) can be integrated to find the general velocity
gradient expression. The viscosity is regarded as a constant.
1
2
1
1
u py c
y x
w py c
y z
(2.22)
where 1c and 2c are constants.
Integrating (2.22) again,
2
1 3
2
2 4
1
2
1
2
p yu c y c
x
p yw c y c
z
(2.23)
where 3c and 4c are constants.
On solving (2.23) while considering the boundary conditions of no slip,
at 0, ,b by u u w w and
at , ,a ay h u u w w
We can find
1( )
2
1( )
2
b a
b a
p h y yu y y h u u
x h h
p h y yw y y h w w
z h h
(2.24)
The Reynolds’ equation can here be created with the help of velocity components u and w
given in the continuity equation (2.20).
1
( ) ...2
1... ( ) 0
2
b a
b a
p h y yy y h u u v
t x x h h y
p h y yy y h w w
z z h h
(2.25)
Bearing theory and governing equations
17
1
( ) ( ) ...2
...
b a
b a
p p h y yv y y h y y h u u
y x x z z x h h
h y yw w
z h h t
(2.26)
Integrating (2.26) with respect to y with the conditions bv v at 0y and av v at y h ,
0 0
1( ) ( ) ...
2
...
h h
a b
b a b a
p pv v y y h dy y y h dy
x x z z
h y y h y yu u w w
x h h z h h t
(2.27)
By using the relation
0 0
( , , ) ( , , ) ( , , )
h hh
f x y z dz f x y z dz f x y hx x x
(2.28)
2 2
0
2 2
0
0
1( ) ( ) ...
2
... ( ) ( ) ....
... 1 1 ...
...
h
a b
y h
h
y h
h
b a b a
y h
p p hv v y yh dy y yh
x x x x
p p hy yh dy y yh
z z z z
y y y y hu u dy u u
x h h h h x
0
1 1
h
b a b a
y h
y y y y hw w dy w w h
z h h h h z t
(2.29)
Reynolds equation for two-dimensional flow
18
3 2 3 2
0 0
2 2
0
2 2
0
1 1...
2 3 2 2 3 2
... ...2 2
...2 2
h h
a b
h
b a a
h
b a a
p y y h p y y hv v
x x z z
y y hy u u u
x h h x
y y hy w w w h
z h h z
t
(2.30)
3 31 1
...2 6 2 6
... ...2 2
...2 2
a b
b a a
b a a
p h p hv v h
t x x z z
h h hu u u
x x
h h hw w w
z z
(2.31)
3 3
...12 12
...2 2
a b
a b a b
a a
h p h pv v h
t x x z z
u u h w w h h hu w
x z x z
(2.32)
3 3
...12 12 2 2
...
a b a b
a b a a
u u h w w hh p h p
x x z z x z
h hv v u w h
x z t
(2.33)
The last four terms written on the right side of (2.33) can be joined and represented as
( )h t . The generalized Reynolds’ equation will then become
3 3
12 12 2 2
a b a bu u h w w h hh p h p
x x z z x z t
(2.34)
Practically, all the components of velocity are not present. Mostly, the following boundary
velocities will concern us,
0,a b a a
hw w v u
x
(2.35)
Bearing theory and governing equations
19
Using (2.35) in (2.34), constant velocities can be obtained
3 3
12 12 2
h hh p h p U
x x z z x t
(2.36)
where a bU u u (2.37)
For steady-state conditions, the generalized Reynolds’ equation (2.36) becomes
3 3
12 12 2
hh p h p U
x x z z x
(2.38)
If the fluid property does not change, as is common for an incompressible lubricant, a
modified 2-D Reynolds’ equation becomes:
3 3 6p p h
h h Ux x z z x
(2.39)
From (2.36), the right side term h t V could help to develop positive pressure. Hence,
when any two surfaces are advancing towards one another, a positive pressure can be
produced. A given amount of finite time is necessary to squeeze the lubricant from the gape.
This process provides an essential; cushioning effect in the bearings. When the two surfaces
are advancing aware from each other, cavitations will probably occur in liquid films.
In case of an isoviscous incompressible fluid, the Reynolds’ equation becomes:
3 3 12p p
h h Vx x z z
(2.40)
2.5.2 Equation for short bearing
In case the bearing is short, the flow caused by the pressure gradient with respect to pressure
variation in the x -direction becomes negligible. Here, the 1-D equation (2.39) becomes
3 6p h
h Uz z x
(2.41)
which is the Reynolds’ equation (Basu et al., 2005; Majumdar, 2008; Shimpi & Deheri, 2010)
when changed while considering the assumptions of general hydrodynamic lubrication.
Reynolds equation for two-dimensional flow
20
The boundary conditions here are 0p at 1
2Z and 0
dp
dZ at 0Z
2.5.3 Equation for infinitely long parallel plates
If it is assumed that the bearing is infinitely long in the axial direction, which implies zero
pressure variation in the z -direction, the p z term in the 2-D Reynolds’ equation can be
avoided. Equation (2.39) in this case will become
3 6d dp dh
h Udx dx dx
(2.42)
With the boundary conditions 1
02
p
2.5.4 Equation for plane slider bearing
When (2.41) is integrated with respect to x , it yields
312 mh hdp
Udx h
(2.43)
where mh is some thickness of the film.
The boundary conditions are (0) 0, (1) 0p p
2.5.5 Equation for parallel circular plate
Consider a case of a circular plate which has a radius of a and is advancing towards a plane
surface parallel to it. For such an axisymmetric case along with polar coordinates, (2.40) will
change to
3112
d dprh V
r dr dr
(2.44)
The boundary conditions are 0p at 0,r a
Bearing theory and governing equations
21
2.5.6 Equation for rectangular plate on a plane surface
A rectangular plate has a normal velocity which is equal to V . When a constant film thickness
of h , is considered, the Reynolds’ equation (2.40) becomes
2 2
2 2 3
12p p V
x z h
(2.45)
The boundary conditions are , 02
ap z
and , 02
bp x
2.5.7 Equation for infinitely long rectangular plate
From (2.45),
2
2 3
12d p V
dz h
(2.46)
The boundary conditions are 02
bp
2.5.8 Equation for complete cone
The differential equation (Prakash & Vij, 1973) is
3
2
1 12
sin
d dp Vxh
x dx dx
(2.47)
The boundary conditions are 0
( cosec ) 0, 0x
dpp a
dx
2.5.9 Equation for truncated cone
The differential equation (Prakash & Vij, 1973) is
3
2
1 12
sin
d dp Vxh
x dx dx
(2.48)
Reynolds equation for two-dimensional flow
22
The boundary conditions are ( cosec ) 0, ( cosec ) 0p a p b
2.5.10 Equation for parallel-step-pad slider bearing
The solutions first consider the regions of inlet and outlet as distinctive and then are combines
at the shared boundary. Hence, the thickness of the film becomes a constant in the two
regions.
From (2.39)
2 2
2 20
p p
x y
(2.49)
The boundary conditions are (0) 0, (1) 0p p
2.5.11 Equation for circular step bearing
Consider an annular ring of length dr at radius r (Majumdar, 2008). The total flow then is
3
212
h dpQ r
dr
(2.50)
The boundary conditions are 0( ) 0, ( )i sp r p r p
2.5.12 Equation for circular disks
From (Deheri & Patel, 2006)
3
1 12
z h
d dp h pr
r dr dr h t z
(2.51)
The boundary conditions are 0
( ) 0, 0, 0, 0r r a z h H
dp dp dpp a
dr dr dz
2.5.13 Equation for Neuringer-Rosensweig model
Neuringer and Rosensweig (1964) proposed an explanation of a magnetic fluid’s steady flow.
Bearing theory and governing equations
23
It was:
Equation of motion
2
0( . ) ( . )p q q q M H (2.52)
Equation of magnetization
M H (2.53)
Equation of continuity
. 0 q (2.54)
Maxwell equations
0 H (2.55)
and
. 0 H M (2.56)
where , , , p q M and are fluid density, fluid velocity, magnetization vector, film pressure
and fluid viscosity respectively.
Also,
ui vj wk q (2.57)
where , ,u v w are components of film fluid velocity in ,x y and z - directions respectively.
Using above (2.53) and (2.55), (2.52) becomes.
2 20.2
p H
q q q (2.58)
This proposes that when a magnetic fluid is used for a lubricant, an additional pressure
2
0 2H is presented in the Navier-Stokes equations. Then, the new Reynolds’ equation
here is derived like (2.36) as
3 2 3 2
0 0
1 16 12
2 2h
hh p H h p H U w
x x z z x
(2.59)
Reynolds equation for two-dimensional flow
24
2.5.14 Equation for Shliomis model
Shliomis (1974) suggested that changing the applied magnetic field can cause a two-way
effect on the particles present in the magnetic fluid. The first possibility is that the particles’
rotation changes or alternatively, their magnetic moment alters. B (Brownian relaxation time
parameter) can be used to find the rotation of the particles and S (relaxation time parameter)
proposes the intrinsic rotation process. If a steady flow is considered and the second
derivatives along with inertial of S are overlooked, the flow equation changes to,
2
0
1. 0
2 s
p I
q M H S (2.60)
1
2 q (2.61)
0 sI S M H (2.62)
0BM
H I
HM S M (2.63)
0 H (2.64)
and
0 H M (2.65)
The following can be derived by combining all the mentioned equations:
2
0 0
1. 0
2p q M H M H
(2.66)
00
B sBM
H I
HM M M H M (2.67)
When the Langevin’s parameter 1 is applied in the case of a strong magnetic field, the
above equation becomes
0B
M
H M H H (2.68)
with
Bearing theory and governing equations
25
6
1 cothBnk T
(2.69)
where
0
0
1coth , Bk T
M n H
(2.70)
For a deferment of spherical particles:
6s
I
and
3B
B
V
k T
(2.71)
further moving with the analysis adopted in (Bhat, 2003). Reynolds’ type equations in case of
Shliomis model with a one dimensional flow when used for an impermeable slider bearing
when the slider is moving with U, a uniform velocity in the x-direction, can be obtained by:
33 33 2 5
0 3
3 312 6
16 320
B Ba a
a a
N Nd dp dh d dp d dph h U U h h
dx dx dx dx dx dx dx
(2.72)
where 0 0 0and
4
Ba
NN M H
2.5.15 Equation for Jenkins model
Jenkins (1972) discussed the model of Ferro-fluid flow. Considering the modifications given
by Maugin, the steady flow model’s equations are (Ram & Verma, 1999):
2
2
0. .2
Ap
M
Mq q q M H q M (2.73)
paired with the equations (2.53)-(2.56) and A as the constant material parameter.
From the equation mentioned above, it can be observed that the Jenkins model represents a
generalization of the Neuringer-Rosensweig model along-with an extra term
2 2
2 2
A A
M H
M Hq M q H (2.74)
Reynolds equation for two-dimensional flow
26
which changes the fluid velocity. Neuringer-Rosensweig changes the pressure and Jenkins
model changes the pressure as well as the velocity of the Magnetic fluid.
Further, with the analysis discussed in (Bhat, 2003). The Generalized Reynolds’ type
equations for the Jenkins model in case of a one dimensional flow when an impermeable slider
bearing is used along-with the slider shifting with U, a uniform velocity in the x-direction, can
be obtained as
320
26 12
21
2
h
d h d dhp H U W
dx dx dxA H
(2.75)
27
CHAPTER 3
Influence of Ferrofluid Lubrication on
Longitudinally Rough Truncated Conical Plates with
Slip Velocity
3.1 Introduction
Various industrial applications including aerospace and aeronautical industries, nuclear and
civil engineering, modern construction engineering amongst others make use of conical plates
as crucial constitutional elements. The dynamic response of these conical plates is
significantly impacted by various fluids (stationary or flowing) that they work with. That is
why, it is crucial to study the behavior generated by different load types in order to ensure safe
functioning in applications. A number of experimental and analytical studies have come
forward recently that study the fluid effects on plates and shells. Flat and curved plates and
circular cylindrical shells have been the major concern for most of them. While there hasn’t
been much work on the fluid effects on conical plates. Thin walled conical plates are used in
many different engineering domains. From aircrafts and satellites in aerospace to submarines,
waterborne ballistic missiles and torpedoes in ocean engineering and containment vessels in
civil, conical shells have a lot of different applications.
Different researchers have come up with a number of ways to study the impact of surface
roughness on bearing performance. Christensen and Tonder (1969a,b, 1970) utilized the
concept of stochastic averaging and created a model with lubricated films having longitudinal
and transverse roughness. Burton (1963), Berthe and Godet (1974) and Gadelmawla et al.
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
28
(2002) used this model and came up with a different geometric configurations to understand
the impact of surface roughness. When two distinct lubricating surfaces generate positive
pressure by approaching each other in normal direction and supports a load, this phenomenon
called the squeeze film. A squeeze film has a lot of applications in automobile and domestic
appliances. That is why, researchers like (Bhat & Deheri, 1991b; Lin et al., 2013c; Prakash &
Vij, 1973; Ting, 1975) worked on studying the impact of a squeeze film bearing. Neuringer
and Rosensweig (1964) designed a basic model of flow to analyze the behavior of Ferrofluid
with different external magnetic fields. A lot of papers have been written that study various
bearing systems under the Neuringer and Rosensweig model, for example circular plates by
(Bhat & Deheri, 1992), (Patel et al., 2012a) in journal bearing and (Patel & Deheri, 2011b) in
plane inclined slider bearing.
Patel and Deheri (2007b) focused their study on analyzing the impact of magnetic fluid on
bearing performance of porous truncated conical plates. Deheri et al. (2007) took their work
further and focused on transverse roughness. They concluded that negative variance and
negative skewness are crucial with appropriate semi vertical angle. Andharia and Deheri
(2011) further worked on this aspect by taking into account the longitudinal roughness.
Shimpi and Deheri (2014b) took into account the slip velocity and deformation impact to
extend the study of (Deheri et al., 2007). Shimpi and Deheri (2016) also studied truncated
conical plates by taking into account longitudinal roughness, slip velocity and deformation.
Vadher et al. (2011) worked on the study by using hydromagnetic bearing instead of
hydrodynamic bearing.
The squeeze films with a magnetic fluid base had an impact on conical plates which has been
studied by a lot of researchers using a number of different parameters. For example, (Patel &
Deheri, 2007a) used porosity, (Patel & Deheri, 2013b) studied the model using transverse
roughness and porosity, (Andharia & Deheri, 2010) worked on longitudinal roughness, (Patel
& Deheri, 2016c) studied it by taking into account longitudinal roughness with slip velocity
and (Patel et al., 2017a) worked with longitudinal roughness as well as deformation effect.
The design of a structure from porous as well as fluid layer is derived by (Beavers & Joseph,
1967). They considered slip boundary condition at the interface. Many researchers have
worked with slip velocity; (Munshi et al., 2017) used circular plates, (Shukla & Deheri, 2013)
worked on Rayleigh step bearing and (Shah & Bhat, 2002) studied inclined slider bearing.
Analysis
29
From these studies, it can be concluded that slip place a crucial part in changing the bearing
capacity of any system.
This study aims to change and clearly define the findings (Andharia & Deheri, 2011) to find
the correlation between slip velocity and Ferrofluid squeeze film in truncated conical plates
with roughness pattern of longitudinal.
3.2 Analysis
The system has two plates in the shape of truncated cones. The upper plate is in motion
towards the lower plate.
The h is considered as
m sh h h (3.1)
We follow the work of (Christensen & Tonder, 1969a,b, 1970) and use the probability density
function
32
2
351 ,
32
0 , elsewhere
ss
s
hc h c
f h c c
(3.2)
FIGURE 3.1 Configuration of truncated conical plates (Andharia & Deheri, 2011)
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
30
In this case, c is the highest possible deviation from the average width of the film. , and
are considered in view of relationships
2 2 3( ), ( ) , ( )s s sE h E h E h (3.3)
In this equation, ( )E represents the expectancy operator which can be calculated by
c
s s
c
E f h dh
(3.4)
Further, the magnetic field’s magnitude is represented by (Andharia & Deheri, 2011)
2 ( cosec )( cosec ),H k a x x b b x a (3.5)
In the equation, k is the suitable constant. If we assume that the magnetic field existing
external has developed as a result of a potential function, ( , )x z making the equation
2 cosec coseccot
2( cosec )( cosec )
x a b
x z a x x b
(3.6)
The general hydrodynamic lubrication assumption modified Reynolds’ (Andharia & Deheri,
2011, Shimpi & Deheri, 2014b) equation as follows:
3 2 00 2
121 1
2 sin
hd dxh p H
x dx dx
(3.7)
The averaging process of stochastic suggest by (Andharia & Deheri, 2011), (3.7) takes the
form
2 003 2
121 1 1
( ) 2 sinm s
hd dx p H
x dx E h h dx
2 00 2
121 1 1
( , , , ) 2 sinm
hd dx p H
x dx g h dx
(3.8)
where
3 1 2 2 2 2 3 3( , , , ) 1 3 6 10 3m m m m mg h h h h h (3.9)
Analysis
31
The following dimensionless quantities are used,
3
0 3
0 0 0 0
3 3
0 0 0
2 2
0 0
, , ( , , , , ) ( , , , ), , , ,
, ,
( ) cosec
mm
hxX h g h s h g h
a h h h h
h k p hbK p
a h h a b
(3.10)
The related pressure limit settings are
( cosec ) 0, (cosec ) 0p K p (3.11)
Solving (3.8) with the aid of (3.11), the non-dimensional type of dispersal of pressure is
2
3 2 2 2
11 sin cosec ...
2 1
... 6 ( , , , , ) cosec 1 sin 2 ln sin
p X X KK
g h s X K X
(3.12)
where
1 1/3 2/3 1
2 2 2 34 4 4 4( , , , , ) 1 3 6( ) 10(3 )
1 1 1 1
s s s sg h s
s s s s
(3.13)
The bearing ability of dimensionless type is obtained
3
0
2 2 2 2 2
0 ( ) cosec
hW W
h a b
(3.14)
3 2
2
52 2 4
2
2 cosec (1 )...
241 cosec
3 ( , , , , ) cosec... 1 1 3 4 ln( )
4 (1 )
KW
K
g h sK K K K
K
(3.15)
where the load bearing capacity is calculated using
cosec
cosec
2
a
b
W xp dx
(3.16)
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
32
3.3 Results and Discussion
As per (3.15), the load bearing ability increase as per the following equation
2 1cosec
12 1
K
K
more than the regular lubricant-based bearing system. In the nonexistence of slip, this
investigation diminishes to the study of (Andharia & Deheri, 2011).
As the (3.15) is linear with regards to , a boost in the magnetization would introduce an
enhancement in the load bearing ability. A comparison of overall performance with (Andharia
& Deheri, 2011) suggests that the impact of slip effect is not all bad. The graphical results are
presented below.
(a) K
(b)
Results and Discussion
33
(c)
(d)
(e)
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
34
(f)
FIGURE 3.2 Profile of W with regards to s
(a)
(b)
Results and Discussion
35
(c)
(d)
FIGURE 3.3 Profile of W with regards to K
(a)
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
36
(b)
(c)
(d) K
FIGURE 3.4 Profile of W with regards to
Results and Discussion
37
(a)
(b)
FIGURE 3.5 Profile of W with regards to
(a)
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
38
(b)
FIGURE 3.6 Profile of W with regards to
FIGURE 3.7 Profile of W with regards to
Following conclusion can be drawn from the above load profiles.
1. There is a substantial increase because of the roughness standard deviation. This
performance increases when there is a high negative skewness value for the surface
roughness.
2. The slip velocity decreases the system’s bearing capacity.
3. The positive impact of magnetization isn’t strong enough to counter the adverse effect
of roughness and slip velocity.
4. However, with an appropriate combination of aspect ratio and semi vertical angle of
the cone, the adverse impact created by surface roughness can be decreased to
significant extent, especially in the case of smaller slip parameter values.
Results and Discussion
39
A comparison of the graphical results presented in (Deheri et al., 2007) goes on to show that
the longitudinally surface roughness can be more adoptable as compare to transverse surface
roughness when no slip is involved.
3.4 Validation
A close scrutiny of the results presented below in tabular form when compared with (Andharia
& Deheri, 2011) suggests that at least 4% enhancement in the load bearing capacity is
registered here. The roughness obstructs the fluid flow, as a result less pressure is generated
but at the same time the magnetization increases the effective viscosity of the lubricant. In
addition, this effect is improved by the negatively skewed roughness. As a result, in this state
the combined positive effect of magnetization and negatively skewed roughness does not
allow the pressure to drop rapidly.
TABLE 3.1 Comparison of W calculated for
Quantity Load carrying Capacity
(calculated for 0.05, 0.3, 0.05, 55 , 0.5, 0.03K s )
Result of the current study Result of Andharia and Deheri (2011)
0.0001 0.2575338 0.2446277
0.001 0.2575457 0.2446395
0.01 0.2576643 0.2447581
0.1 0.2588503 0.2459441
1 0.2707102 0.2578040
TABLE 3.2 Comparison of W calculated for
Quantity Load carrying Capacity
(calculated for 0.01, 0.3, 0.05, 55 , 0.5, 0.03K s )
Result of the current study Result of Andharia and Deheri (2011)
-0.05 0.2576640 0.2447581
-0.02 0.2432010 0.2298273
0 0.2340100 0.2202664
0.02 0.2251560 0.2109807
0.05 0.2124570 0.1974948
TABLE 3.3 Comparison of W calculated for
Quantity Load carrying Capacity
(calculated for 0.01, 0.05, 0.05, 55 , 0.5, 0.03K s )
Result of the current study Result of Andharia and Deheri (2011)
0.1 0.2187241 0.2088460
0.2 0.2333267 0.2223130
0.3 0.2576643 0.2447581
Influence of ferrofluid lubrication on longitudinally rough truncated conical plates with slip
velocity
40
0.4 0.2917369 0.2761812
0.5 0.3355446 0.3165824
TABLE 3.4 Comparison of W calculated for
Quantity Load carrying Capacity
(calculated for 0.01, 0.05, 0.3, 55 , 0.5, 0.03K s )
Result of the current study Result of Andharia and Deheri (2011)
-0.05 0.2576643 0.2447581
-0.02 0.2443342 0.2237814
0 0.2354474 0.2097969
0.02 0.2265606 0.1958125
0.05 0.2132305 0.1748358
TABLE 3.5 Comparison of W calculated for
Quantity Load carrying Capacity
(calculated for 0.01, 0.05, 0.3, 0.05, 0.7, 0.03K s )
Result of the current study Result of Andharia and Deheri (2011)
40° 0.6794927 0.6454508
45° 0.4641394 0.4408880
50° 0.3369186 0.3200416
55° 0.2576643 0.2447581
60° 0.2062889 0.1959567
TABLE 3.6 Comparison of W calculated for K
Quantity Load carrying Capacity
(calculated for 0.01, 0.05, 0.3, 0.05, 55 , 0.03s )
K Result of the current study Result of Andharia and Deheri (2011)
0.1 0.3941064 0.3743720
0.2 0.3728085 0.3541385
0.3 0.3412229 0.3241333
0.4 0.3021199 0.2869877
0.5 0.2576643 0.2447581
Further the effect of variance is sharper in comparison with the investigation of (Andharia &
Deheri, 2011). The consolidated effect of and are akin to that of (Andharia & Deheri,
2011). However the effect of semi vertical angle surges ahead and the negative impact of
roughness displays more variation in this situation despite the fact that standard deviation
raises the load bearing capacity.
3.5 Conclusions
From this study, it can be concluded that appropriate magnetic strength can counter the slip
effect in the case of small order of roughness. Thus, for industrial applications using such a
system is more appropriate, especially if the slip is at the lowest level.
Conclusions
41
Considering the life period perspective, this study is beneficial sine it aids the process of
choosing the ideal aspect ratio, angle. Such an angle can, in turn, reduce the negative effects of
roughness slip combine, even for moderate magnetic field.
42
CHAPTER 4
Effect of Slip Velocity on a Ferrofluid Based
Longitudinally Rough Porous Plane Slider Bearing
4.1 Introduction
The slider bearings are primarily created to aid the transverse load in any given engineering
system. The plane slider bearing study is a classical one. Plane slider bearing has a lot of
applications in various fields including domestic appliances, automobile transmissions and
clutch plates. Murti (1974), Patel and Gupta (1983), Tichy and Chen (1985), Patel et al. (2014)
and Patel et al. (2015a) have also carried out research works on slider bearings.
More and more studies have been recently carried out that analyze the relationship between
surface roughness and the associated hydrodynamic lubrication for a variety of bearing
systems. The reason for this is that practically, all surfaces contain a certain level of
roughness. This may be further exaggerated by some wear and tear. Many researchers have
studied the impact of roughness on load carrying capacity of a system (Andharia et al., 1997,
2000; Chiang et al., 2005; Tzeng & Saibel, 1967). Christensen and Tonder (1969a,b, 1970)
gave a general study that analyzed longitudinal and transverse roughness. This method has
been further used in many different ways in a number of different investigations (Deheri et al.,
2004; Deheri et al., 2013; Panchal et al., 2016; Patel & Deheri, 2011; Patel & Deheri, 2016c).
Contemporary researchers are consistently focusing on studying the magnetic fluid
lubrication, in theory as well as in practice (Shukla & Kumar, 1987). Minute magnetic gains
covered with surfactants are suspended and then dispersed into solvents like kerosene,
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
43
fluorocarbons, hydrocarbons, etc. that are non-conducting and magnetically passive in nature
to create magnetic fluid lubrication. Other researchers including (Andharia et al., 2001; Bagci
& Singh, 1983; Hamrock, 1994; Pinkus & Sternlicht, 1961) have studied hydrodynamic
lubrication with a variety of film shapes.
Furthermore, porosity was introduced in an attempt to decrease the friction. Porous bearings
are used in horsepower motors of hair dryers, record players, vacuum cleaners, tape recorders,
sewing machines, water pumps, etc. Morgan and Cameron (1957) were the first investigators
to study the hydrodynamic lubrication theory of bearings with porous structure.
All the particles undergo a body force when subjected to a magnetic field, resulting in the drag
to flow. Therefore, for industrial application, the study of porous metal lubrication with
magnetic fluid is of primary importance (Patel et al., 2012a). Some researchers (Bhat &
Deheri, 1991a; Neuringer & Rosensweig, 1964; Shah & Bhat, 2002; Shimpi & Deheri, 2014a;
Snyder, 1962) have also used magnetic fluid as a lubricant in order to aid the tribological
performance of a sliding interface. Lin (2013) studied the load carrying capacity of a bearing
system by replacing the lubricant with a magnetic fluid. All these studies had similar
conclusions mentioning that a magnetic fluid, when used as a lubricant, enhances the bearing
system’s performance.
A lot of studies have attempted to explore the impact of slip on different bearings in a
theoretical as well as experimental manner (Munshi et al., 2017; Patel & Deheri, 2011a; Patel
& Deheri, 2013c; Shukla & Deheri, 2013; Sparrow et al., 1972). All these investigations have
concluded that slip has a substantial impact on the working of any bearing system. Andharia
and Deheri (2014) concluded that standard deviation in longitudinal roughness is crucial for
increasing the load carrying capacity. Thus, it was understood that the study of plane slider
bearing having magnetic fluid lubrication should always be assisted with surface roughness
for precise results.
In the work mentioned above, a no slip condition has been taken into account. That is why,
this study use magnetic fluid for lubricant along with calculation of load in terms of magnetic
parameter, roughness parameters and slip parameter. Due to this reason, the configuration of
(Andharia & Deheri, 2014) has been used to investigate the impacts of slip velocity and
porosity.
Analysis
44
4.2 Analysis
Fig. 4.1 displays the bearing configuration that is considered to be infinite on the Y-axis. The
X-axis represents the uniform velocity U of the slider. The minimum and maximum film
thicknesses are represented by 2h and 1h respectively. L is the bearing length.
FIGURE 4.1 Physical geometry of the bearing system (Andharia & Deheri, 2014)
The mentioned field of magnetism is thought to be sloping against the stator, as suggested by
(Andharia & Deheri, 2014). The h is believed to be
m sh h h (4.1)
using the works of (Christensen & Tonder, 1969a,b, 1970). Also, the study uses sh using the
probability density function
32
2
351 ,
32
0 , elsewhere
ss
s
hc h c
f h c c
(4.2)
In this case, c is the highest possible deviation from the average width of the film.
, and are considered by:
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
45
2 2 3( ), ( ) , ( )s s sE h E h E h (4.3)
Here, ( )E represents the anticipated value as provided by
c
s s
c
E f h dh
(4.4)
Neuringer and Rosensweig (1964) devised a theory that describes the stable movement of
magnetic fluid. The model was as follows:
Equation of motion
2
0( . ) ( . )p q q q M H (4.5)
Equation of magnetization
M H (4.6)
Continuity equation
. 0 q (4.7)
The formulae given by Maxwell
0 H (4.8)
and
. 0 H M (4.9)
where , , , p q M and are fluid density, fluid velocity, magnetization vector, film pressure
and fluid viscosity respectively.
Also,
ui vj wk q (4.10)
Further, the magnetic field’s magnitude is given by
2 ( )H kx L x (4.11)
Analysis
46
where k is suitable constant. The general hydrodynamic lubrication assumption modified
Reynolds’ (Deheri et al., 2004; Panchal et al., 2016; Andharia & Deheri, 2014) equation as
follows:
3 2
0
16
2
d d dhh p H U
dx dx dx
(4.12)
Following the stochastically average process discussed in (Andharia & Deheri, 2014), (4.12)
takes the form
2
03 1
1 1 16
( ) 2 ( )m s m s
d d dp H U
dx E h h dx dx E h h
2
0
1 2
1 1 16
( , , , , ) 2 ( , , , , )m m
d d dp H U
dx g h K dx dx g h K
(4.13)
where
3 1 2 2 2 2 3 3
1( , , , , ) 1 3 6 10 3 12m m m m mg h K h h h KH h
(4.14)
1 1 2 2 2 2 3 3
2( , , , , ) 1 3 12m m m m mg h K h h h KH h
(4.15)
The following dimensionless quantities are used,
3
1 2 1 2 2 2
2 2
0 2 2
3 3
2 2 2 2 2 2
, ( , , , , , ) ( , , , , ), ( , , , , , ) ( , , , , ),
, , , , , , ,2
m m
m
xX g h s h g h K g h s h g h K
L
h k h L p hQ KHh Q p
h h h h h h U U L
(4.16)
The associated boundary conditions are
(0) 0, (1) 0p p (4.17)
Solving (4.13) with the aid of (4.17), the non-dimensional type of dispersal of pressure is
2
1
0 2
1( ) 6 ( , , , , , )
( , , , , , )
X
p X X g h s Q dXg h s
(4.18)
where
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
47
1 1/3 2/3
2 2
1
1
2 3
4 4 4( , , , , , ) 1 3 6( ) ...
1 1 1
4... 10(3 12 ) ,
1
sh sh shg h s
sh sh sh
sh
sh
(4.19)
1/3 1/3 2/3
2 2
2
1
2 3
4 4 4( , , , , , ) 1 ( ) ...
1 1 1
4... (3 12 ) ,
1
sh sh shg h s
sh sh sh
sh
sh
(4.20)
and
1
1
20
1
1
0
( , , , , , )
( , , , , , )
( , , , , , )
g h sdX
g h sQ
g h s dX
(4.21)
The load carrying capacity in dimensionless form is obtained
2
2
2
hW W
U L (4.22)
1
11
0 2
6 ( , , , , , ) (1 )6 ( , , , , , ) (1 )
6 ( , , , , , )
g h s XW Q g h s X dX
g h s
(4.23)
where the load carrying capacity is calculated using
1
0
W p dx (4.24)
4.3 Results and Discussion
The study shows that (4.18) and (4.23) display the dimensionless pressure distribution and
dimensionless load carrying capacity respectively. Andharia and Deheri (2014) suggested the
equation of film thickness h in terms of 2 X to study the porous plane slider bearing. The
Results and Discussion
48
results are used to analyze the correlation of slip velocity and longitudinally rough porous
plane slider bearing with magnetic fluid.
1/3 rule as given by Simpson having a step size of 0.2 is helpful in calculating (4.23) for
altering the measure of , porosity , roughness parameters , , and slip parameter s .
Figs. 4.2-4.6 represent these results graphically.
The variation in the W with regards to for different values of , , , and s is shown in
the Figs. 4.2(a to e). It can be concluded from these figures that, using magnetic fluid
lubrication substantially increases the bearing performance. Moreover, the load-bearing
capacity is directly proportional to the magnetization. From the physical point of view
magnetization increases the viscosity of the lubricant which enhances pressure and
consequently the load-bearing capacity. The impact of standard deviation due to
magnetization of this performance is almost marginal. Figs. 4.3(a to d) suggest that with
positive variance, the W decreases while with negative variance, it increases. Figs. 4.4(a to c)
suggest that W is positively related to . It is observed that the large values of may lead to
subjection of bearing surfaces particularly in the situation when the bearing is operating in the
boundary lubrication regime. From Figs. 4.5(a to b) it can be said that negative skewed
roughness has a positive impact on the W . As a result, in this state the combined positive
effect of magnetization and negatively skewed roughness does not allow the pressure to drop
rapidly. The total impact of (+ve), (+ve) and porosity are important as they can severely
decrease the W . Thus, it can be concluded that the slip velocity has a negative impact on the
bearing performance (Fig. 4.6) therefore the role of slip velocity is to decrease the resistance
encountered by fluid flowing in the gap itself and, by this means, to diminish the load-carrying
capacity.
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
49
(a)
(b)
(c)
Results and Discussion
50
(d)
(e) s
FIGURE 4.2 Profile of W with regards to
(a)
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
51
(b)
(c)
(d) s
FIGURE 4.3 Profile of W with regards to
Results and Discussion
52
(a)
(b)
(c) s
FIGURE 4.4 Profile of W with regards to
Effect of slip velocity on a ferrofluid based longitudinally rough porous plane slider bearing
53
(a)
(b) s
FIGURE 4.5 Profile of W with regards to
FIGURE 4.6 Profile of W with regards to
Conclusions
54
4.4 Conclusions
It can be said that the attempts made to neutralize the adverse impacts of surface roughness,
porosity and slip velocity with the help of magnetization are considerably limited. Thus,
surface roughness of a bearing system should be given some special attention during designing
of the system even if the slip velocity is kept minimum. In particular, a substantial gain in
response can be attained by the selection of porous materials which accentuate slip velocity.
However, one thing that remains consistent is that for a better bearing performance, the slip
velocity should always be minimum.
55
CHAPTER 5
Numerical Modelling of Shliomis Model Based
Ferrofluid Lubrication Performance in Rough Short
Bearing
5.1 Introduction
The slider bearing is one of the most basic and commonly used hydrodynamic bearing. The
simplicity of the film thickness expression and the straightforwardness of boundary conditions
can be held accountable for the same. Unlike other bearings, slider bearings do not create
negative pressure which can be problematic for load bearing. This is because their film is
continuous and non-diverging. Thus, they support axial loads. Many researchers have studied
non-porous sliders. Christensen and Tonder (1969a,b, 1970) used a stochastic concept and
came up with a new model for lubricated surfaces with striated roughness using an averaging
film. They derived stochastic Reynolds’ equation and used the results to study the impact of
surface roughness on the load bearing capacity in a rough bearing system. Shliomis (1974)
analyzed the modes of creating magnetic colloids along with the stability concerns. Patel et al.
(2010a) worked on studying the performance of a smooth short bearing. Deheri and Patel
(2011) and Patel et al. (2010b) analyzed the performance of a short rough bearing with a zero
mean. They worked with a variety of magnetic field magnitudes in their study. Shimpi and
Deheri (2010) further worked on the results of (Deheri & Patel, 2011; Patel et al., 2010b)
focusing on short bearings with the non-zero mean with a different form of the magnetic field
magnitude. Shimpi and Deheri (2012b) worked on it by including a deformation effect as well.
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
56
Patel and Deheri (2013a) studied further comparing two different types of porous structures.
Patel and Deheri (2013c) extended the study of (Patel et al., 2010a) by adding the aspect of
slip velocity. They concluded that optimal performance can only be achieved at the minimum
slip. It should be noted that the researches mentioned above used the Neuringer-Rosensweig
model. Patel et al. (2015b) conducted a different study by replacing the hydrodynamic bearing
of the above-mentioned studies with a hydromagnetic porous short bearing. Additionally, it is
a commonly known fact that roughness impacts the load carrying capacity substantially.
By reviewing literature concerning Ferrofluid flow, one can understand that Shliomis model
displays better results than Neuringer-Rosensweig model. Thus, this study is focused on
scrutinizing the performance of Ferrofluid lubrication using a short bearing based on the
Shliomis model.
5.2 Analysis
The configuration of the short bearing system (infinitely short in the z -direction) is presented
in Fig. 5.1. The slider has a velocity U in the x -direction. The breadth B is in the z -
direction where B L (length). The pressure gradient p x can be neglected as the
pressure gradient p z remains much larger.
FIGURE 5.1 Configuration of the bearing system (Patel & Deheri, 2013a)
The thickness h is considered as
m sh h h (5.1)
where mh is taken as (Patel & Deheri, 2013a):
Analysis
57
1 22
2
1 1 ,m
h hxh h m m
L h
We follow the works of (Christensen & Tonder, 1969a,b, 1970) and use the probability
density function (Christensen & Tonder, 1969a,b, 1970)
32
2
351 ,
32
0 , elsewhere
ss
s
hc h c
f h c c
(5.2)
with c being the maximum deviation from the mean film thickness. , and are given
by the relationships (Christensen & Tonder, 1969a,b, 1970)
2 2 3( ), ( ) , ( )s s sE h E h E h (5.3)
where ( )E denotes the expectancy operator given by
( ) ( ) ( )
c
s s
c
E f h dh
(5.4)
Actually, magnetic fluids or Ferrofluids have a constant nature and are a type of colloidal
suspensions that possess extremely superior magnetic particles in a viscous fluid. We can use
an external magnetic field to position, limit or monitor these fluids as required. This, in turn,
increases the fluid effective viscosity. This research has substantially contributed to an
increased application of these magnetic fluids in bearing systems as lubricating agents. A
noteworthy fact here is that major studies in the field use the Neuringer-Rosensweig model
suggesting that magnetization vector is parallel to the applied magnetic field. Since the
Shliomis model considers particle rotation, it overcomes this limitation.
Shliomis (1972, 1974) proposed that a change in the applied magnetic field can have a two-
way implication on the particles in the magnetic fluid. Either, rotation of such particles is
impacted or the magnetic moment changes in them. B (Brownian relaxation time parameter)
is used to derive the particle rotation and S (magnetic moment relaxation time parameter)
suggests the intrinsic process of rotation. Considering a steady flow while overlooking the
inertial and second derivatives of S , the equations of flow becomes,
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
58
2
0
1. 0
2 s
p I
q M H S (5.5)
where
1
2 q (5.6)
0 sI S M H (5.7)
0BM
H I
HM S M (5.8)
0 H (5.9)
and
0 H M (5.10)
The following can be derived by combining all the above mentioned equations; in the light of
the procedure given in (Shliomis, 1974)
2
0 0
1. 0
2p q M H M H
(5.11)
00
B sBM
H I
HM M M H M (5.12)
Langevin’s parameter 1 is used for a strong magnetic field, the above equation (Bhat,
2003; Shliomis, 1972, 1974) changes to
0B
M
H M H H (5.13)
with
6
1 cothBnk T
(5.14)
where
Analysis
59
0
0
1coth , Bk T
M n H
(5.15)
For a deferment of spherical particles:
6s
I
and
3B
B
V
k T
(5.16)
In view of the discussion of (Shliomis, 1974) the flow remains in the xz - plane while the
magnetic field is taken in the y -direction by making use of the assumptions for ( , , )U u v w
and ( , , )x y zH H HH where in ,u w v and ,y x zH H H . In the light of the boundary
conditions of the magnetic field components at the plates, it can be inferred that 0yH H
while ,x zH H remain negligible in comparison with 0H . Therefore, for the axially symmetric
flow, the associated uniform magnetic field may be represented by 00, ,0HH .
Equations (5.11) to (5.13) develop into the following (Majumdar, 2008)
1
0
1
p u
x y y
p
y
p w
z y y
(5.17)
and
( ) ( ) ( ) 0u v wt x y z
(5.18)
where (referring to Shliomis, 1972)
3 tanh
,2 tanh
(5.19)
and
0
51
2
(5.20)
Solving (5.17) under no slip boundary conditions (Majumdar, 2008);
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
60
at 0: ,b by u u w w
at : ,a ay h u u w w
one can find
1( )
2 (1 )
1( )
2 (1 )
b a
b a
p h y yu y y h u u
x h h
p h y yw y y h w w
z h h
(5.21)
Putting the velocity components expression into continuity equation (5.18) and integrating
under the conditions bv v at 0y and
av v at y h gives rise to;
0 0
0 0
1( ) ( )
2 (1 ) (1 )
h h
a b
h h
b a b a
p pv v y y h dy y y h dy
x x z z
h y y h y yu u dy w w dy
x h h z h h
ht
(5.22)
In reference to the experiment through the utilization of relation (Majumdar, 2008)
0 0
( , , ) ( , , ) ( , , )
h hh
f x y z dz f x y z dz f x y hx x x
(5.23)
one obtains for constant velocities
3 3
12 (1 ) 12 (1 ) 2
h p h p Uh h
x x z z x t
(5.24)
where
a bU u u (5.25)
Generalized Reynolds’ equation (5.24) turns in the state of equilibrium, which brings
(Majumdar, 2008)
3 3
12 (1 ) 12 (1 ) 2
h p h p Uh
x x z z x
(5.26)
Analysis
61
Modified two dimensional Reynolds’ equation for an incompressible lubricant is
3 3
12 (1 ) 12 (1 ) 2
h p h p U h
x x z z x
(5.27)
In the x -direction, the flow because of pressure gradient in the variation of pressure can be
avoided when the bearing is short. In this case, one dimensional equation (5.27) leads to
3 6 1p h
h Uz z x
(5.28)
which is Reynolds’ equation (Basu et al., 2005; Majumdar, 2008; Shimpi & Deheri, 2010)
modified according to the general hydrodynamic lubrication assumptions.
According to the stochastically average process (Shimpi & Deheri, 2010), (5.28) becomes:
3 6 1p
E h U E hz z x
, , , 6 1m m
pg h U h
z z x
(5.29)
where
3 2 2 2 2 3, , , 3 3 3m m m mg h h h h (5.30)
The following dimensionless quantities are used,
3
2 2
3
2
3 2
2 2 2 0 2 2
, , ,, , 1 1 , , , , ,
, , , , ,
mmg hhx z
X Z h m X g hL B h h
ph L Bp L B
h h h UB h h
(5.31)
The associated boundary conditions (Deheri & Patel, 2011; Lin et al., 2013b; Patel et al.,
2012a) are
0p at 1
2Z and
d0
d
p
Z at 0Z (5.32)
With the aid of (5.32), the pressure distribution in a non-dimensional form comes out to be
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
62
2
3 1 2.5 1 1
4, , ,
mp Z
L g h
(5.33)
where
3 2 2 2 2 3
, , , 3 3 3g h h h h (5.34)
The load bearing capacity in the dimensionless form (Patel & Deheri, 2013a) is obtained
3
2
4
0
h WW
UB (5.35)
1
0
1 2.5 1 d
2 , , ,
m XW
B g h
(5.36)
where the load bearing capacity is calculated using (Patel & Deheri, 2013a)
2
0
2
, d d
B
L
B
W p x z x z
(5.37)
5.3 Results and Discussion
Expressions (5.33) and (5.36) that can signify a dimensionless form of pressure and bearing
load carrying capacity are found using Reynolds’ equation. The load carrying capacity
equation (5.36) is then solved numerically with the help of Simpson’s 1/3 rule to analyze the
impact on the bearing system. From the graphical representation, it can be concluded that
ferrofluid lubrication based on the Shliomis model can significantly neutralize the negative
effects of the bearing roughness on its load carrying capacity.
The final values of W using different parameters are plotted graphically. The Figs. 5.2(a-f)
display the changes in W corresponding to different values of . They suggest that
magnetization leads to a substantial increase in W . This may be probably due to the fact that
the magnetization increases the effective viscosity of the lubricant there by increasing the
pressure. It is noticed from Fig. 5.2e that the value of maximum W derived is 0.094 at a
Analysis
63
smaller value of 0.05 with regards to . From Figs. 5.3(a-e), it can be suggested that
with an increase in W , shifts from 0.2 to 1. It may be desirable to evaluate exclusively the
contribution of the volume concentration parameter, for enhancing the bearing performance.
Figs. 5.4(a-d) display the impact of the aspect ratio on W and suggests that the aspect ratio
causes a sharp increase in W . The decrease in the value of W in Figs. 5.5(a-c) display the
unfavorable impact of standard deviation on the bearing system performance. However, this
investigation exhibits the unfavorable standard deviation associated with the roughness which
could be neutralized up to certain extent by the positive effect of the magnetization parameter,
by suitably choosing film thickness ratio. Figs. 5.6(a-b) establish an inverse relationship
between the variance and W suggesting that positive variance reduces W while negative
variance increases it. The decrease in the load carrying capacity is basically due to the fact that
transverse roughness retards the motion of the lubricant. Under the impact of skewness, W
changes according to Fig. 5.7.
(a)
0.015
0.024
0.033
0.042
0.051
0.1 0.2 0.3 0.4 0.5
W̅
τ
υ = 0.2
υ = 0.4
υ = 0.6
υ = 0.8
υ = 1
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
64
(b) m
(c)
(d)
0.010
0.030
0.050
0.070
0.090
0.1 0.2 0.3 0.4 0.5
W̅
τ
m = 0.1
m = 0.3
m = 0.5
m = 0.7
m = 0.9
0.040
0.055
0.070
0.085
0.100
0.1 0.2 0.3 0.4 0.5
W̅
τ
σ̅ = 0.1
σ̅ = 0.2
σ̅ = 0.3
σ̅ = 0.4
σ̅ = 0.5
0.050
0.062
0.074
0.086
0.098
0.1 0.2 0.3 0.4 0.5
W̅
τ
α̅ = -0.05
α̅ = -0.02
α̅ = 0
α̅ = 0.02
α̅ = 0.05
Results and Discussion
65
(e)
(f) B
FIGURE 5.2 Profile of W with regards to
(a) m
0.060
0.070
0.080
0.090
0.100
0.1 0.2 0.3 0.4 0.5
W̅
τ
ε̅ = -0.05
ε̅ = -0.02
ε̅ = 0
ε̅ = 0.02
ε̅ = 0.05
0.010
0.030
0.050
0.070
0.090
0.1 0.2 0.3 0.4 0.5
W̅
τ
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
0.004
0.016
0.028
0.040
0.052
0.2 0.4 0.6 0.8 1
W̅
φ
m = 0.1
m = 0.3
m = 0.5
m = 0.7
m = 0.9
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
66
(b)
(c)
(d)
0.020
0.032
0.044
0.056
0.068
0.2 0.4 0.6 0.8 1
W̅
φ
σ̅ = 0.1
σ̅ = 0.2
σ̅ = 0.3
σ̅ = 0.4
σ̅ = 0.5
0.020
0.035
0.050
0.065
0.080
0.2 0.4 0.6 0.8 1
W̅
φ
α̅ = -0.05
α̅ = -0.02
α̅ = 0
α̅ = 0.02
α̅ = 0.05
0.030
0.045
0.060
0.075
0.090
0.2 0.4 0.6 0.8 1
W̅
φ
ε̅ = -0.05
ε̅ = -0.02
ε̅ = 0
ε̅ = 0.02
ε̅ = 0.05
Results and Discussion
67
(e) B
FIGURE 5.3 Profile of W with regards to
(a)
(b)
0.000
0.020
0.040
0.060
0.080
0.2 0.4 0.6 0.8 1
W̅
φ
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
0.010
0.025
0.040
0.055
0.070
0.1 0.3 0.5 0.7 0.9
W̅
m
σ̅ = 0.1
σ̅ = 0.2
σ̅ = 0.3
σ̅ = 0.4
σ̅ = 0.5
0.010
0.025
0.040
0.055
0.070
0.085
0.1 0.3 0.5 0.7 0.9
W̅
m
α̅ = -0.05
α̅ = -0.02
α̅ = 0
α̅ = 0.02
α̅ = 0.05
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
68
(c)
(d) B
FIGURE 5.4 Profile of W with regards to m
(a)
0.010
0.030
0.050
0.070
0.090
0.1 0.3 0.5 0.7 0.9
W̅
m
ε̅ = -0.05
ε̅ = -0.02
ε̅ = 0
ε̅ = 0.02
ε̅ = 0.05
0.001
0.021
0.041
0.061
0.081
0.1 0.3 0.5 0.7 0.9
W̅
m
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
0.027
0.034
0.041
0.048
0.055
0.1 0.2 0.3 0.4 0.5
W̅
σ̅
α̅ = -0.05
α̅ = -0.02
α̅ = 0
α̅ = 0.02
α̅ = 0.05
Results and Discussion
69
(b)
(c) B
FIGURE 5.5 Profile of W with regards to
(a)
0.032
0.038
0.044
0.050
0.056
0.1 0.2 0.3 0.4 0.5
W̅
σ̅
ε̅ = -0.05
ε̅ = -0.02
ε̅ = 0
ε̅ = 0.02
ε̅ = 0.05
0.005
0.017
0.029
0.041
0.053
0.1 0.2 0.3 0.4 0.5
W̅
σ̅
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
0.034
0.038
0.041
0.045
0.048
-0.050 -0.025 0.000 0.025 0.050
W̅
α̅
ε̅ = -0.05
ε̅ = -0.02
ε̅ = 0
ε̅ = 0.02
ε̅ = 0.05
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
70
(b) B
FIGURE 5.6 Profile of W with regards to
FIGURE 5.7 Profile of W with regards to
5.4 Validation
The validation of the conclusion of this paper has been achieved by using the following
comparison sets used in other publication. Tables 5.1 to 5.5 depict the increase in load
carrying capacity over 14%.
TABLE 5.1 Comparison of W calculated for
Quantity Load bearing Capacity (calculated for
0.09, 0.1, 0.05, 0.1, 0.1, 0.3, 10, 0.5m B L )
Result of the current study Result of Patel and Deheri(2013a) increase in %
-0.05 0.0148500 0.0115134 28.98
0.004
0.014
0.024
0.034
0.044
-0.050 -0.025 0.000 0.025 0.050
W̅
α̅
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
0.001
0.009
0.017
0.025
0.033
-0.050 -0.025 0.000 0.025 0.050
W̅
ε̅
B̅ =10
B̅ =20
B̅ =30
B̅ =40
B̅ =50
Validation
71
-0.02 0.0136554 0.0106446 28.28
0 0.0129308 0.0101176 27.80
0.02 0.0122575 0.0096279 27.31
0.05 0.0113339 0.0089562 26.54
TABLE 5.2 Comparison of W calculated for
Quantity Load bearing Capacity (calculated for
0.09, 0.02, 0.05, 0.1, 0.1, 0.3, 10, 0.5m B L )
Result of the current study Result of Patel and Deheri(2013a) increase in %
0.1 0.0122575 0.0096279 27.31
0.2 0.0114705 0.0090555 26.66
0.3 0.0103649 0.0082515 25.61
0.4 0.0091364 0.0073581 24.16
0.5 0.0079318 0.0064819 22.36
TABLE 5.3 Comparison of W calculated for
Quantity Load bearing Capacity (calculated for
0.09, 0.02, 0.1, 0.1, 0.1, 0.3, 10, 0.5m B L )
Result of the current study Result of Patel and Deheri(2013a) increase in %
-0.05 0.0122575 0.0096279 27.31
-0.02 0.0120166 0.0094527 27.12
0 0.0118614 0.0093399 26.99
0.02 0.0117104 0.0092300 26.87
0.05 0.0114912 0.0090706 26.68
TABLE 5.4 Comparison of W calculated for m
Quantity Load bearing Capacity (calculated for
0.09, 0.02, 0.1, 0.02, 0.1, 0.1, 10, 0.5B L )
m Result of the current study Result of Patel and Deheri(2013a) increase in %
0.1 0.0050043 0.0043529 14.96
0.3 0.0117104 0.0092300 26.87
0.5 0.0158152 0.0122153 29.46
0.7 0.0184708 0.0141467 30.56
0.9 0.0202714 0.0154562 31.15
TABLE 5.5 Comparison of W calculated for B
Quantity Load bearing Capacity (calculated for
0.09, 0.02, 0.1, 0.02, 0.1, 0.1, 0.3, 0.5m L )
B Result of the current study Result of Patel and Deheri(2013a) increase in %
10 0.0117104 0.009230 26.87
20 0.0058552 0.0046150 26.87
30 0.0039035 0.0030767 26.87
40 0.0029276 0.0023075 26.87
50 0.0023421 0.0018460 26.87
Numerical modelling of Shliomis model based ferrofluid lubrication performance in rough
short bearing
72
5.5 Conclusions
The impact of Ferrofluid lubrication on the load bearing capacity of a short bearing system
with a rough surface is studied. From the numerical computations performed, the analysis has
yielded the following conclusions:
• Shliomis’ Ferrofluid flow provides relevant insights on the impact of rotations of the
career liquid and magnetic particles. Furthermore, a varying magnetic field provides
the benefit of creating the maximum field according to the necessary contact area of
the bearing.
• This work is crucial because it provides more freedom than (Verma, 1986) and
(Prajapati, 1994) regarding the magnitude.
• Standard deviation is the most important parameter in determining the performance of
a bearing system of this type.
• Negatively skewed roughness aids the load carrying capacity and boosts the
performance. Another thing to be understood is that a constant magnetic field shows a
positive effect on the bearing capacity in the Shliomis model while the same is not true
for Neuringer-Rosensweig Ferrofluid flow model.
• Furthermore, this article can create a new pathway for ensuring maximum utilization
of the bearing system. It also clearly proposes that by managing the lubricant loss, the
life span of the load bearing system can be increased substantially.
73
CHAPTER 6
Lubrication of Rough Short Bearing on Shliomis
Model by Ferrofluid Considering Viscosity Variation
Effect
6.1 Introduction
Analytical studies performed on hydrodynamic lubrication in a short-bearing non-porous
system are very popular. Many famous books of the field (Bhat, 2003; Hamrock, 1994;
Majumdar, 2008) discuss the Reynolds’ equation and try to derive an exact solution to it by
using different basic film geometries.
All the particles undergo a body force when subjected to a magnetic field, resulting in the drag
to flow. Therefore, for industrial application, the study of lubrication with magnetic fluid is of
primary importance. Some researchers (Deheri et al., 2016; Munshi et al., 2017; Munshi et al.,
2020; Patel et al., 2017c; Patel et al., 2020a,b; Patel et al., 2020c; Vashi et al., 2018) have also
used magnetic fluid as a lubricant in order to aid the tribological performance of a sliding
interface.
Christensen and Tonder (1969a,b, 1970) used a stochastic concept and came up with a new
model for lubricated surfaces with striated roughness using an averaging film. They derived
the stochastic Reynolds’ equation and used the results to study the impact of surface
roughness on the load bearing capacity in a rough bearing system. Shliomis (1974) analyzed
the modes of creating magnetic colloids along with the stability concerns. Patel et al. (2010a)
studied the efficiency and effectiveness of a short bearing with a smooth surface. Deheri and
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
74
Patel (2011) and Patel et al. (2010b) studied the effectiveness of a short bearing with
roughness having a mean zero. They worked with a variety of magnetic field magnitudes for
their study. Shimpi and Deheri (2010) further worked on the results of (Deheri & Patel, 2011;
Patel et al., 2010b) focusing on short bearings with non-zero mean with different form of
magnetic field magnitude. Shimpi and Deheri (2012b) worked on it by including a
deformation effect as well. Patel and Deheri (2013a) further studied comparing two different
types of porous structures. Patel and Deheri (2013c) extended the study of (Patel et al., 2010a)
by adding the aspect of slip velocity. They concluded that optimal performance can only be
achieved at minimum slip. It should be noted that the researches mentioned above used the
Neuringer-Rosensweig model. Patel et al. (2015b) conducted a different study by replacing the
hydrodynamic bearing of the above-mentioned studies with hydromagnetic porous short
bearing. Additionally, it is a commonly known fact that roughness impacts the load bearing
capacity substantially.
Transformation of heat problem has been investigated with non-Newtonian fluid by (Singh et
al., 2018). The positive and negative wall inclination design chart, demonstrate through
(Gupta et al., 2019). Ramadevi et al. (2018) introduced a new arena in the respective field
where the viscosity of fluid and partial slip velocity involved. The payload capacity of
amphibious vehicle with 7kg conceptual designed by (Gokul et al., 2019). Khamari et al.
(2019) represented the impact toughness and microstructure comparison.
A lot of research in the field of Tribology today also focuses on the impact caused by
hydrodynamic lubrication. Most of the studies take viscosity to be a constant value even
though it is a function of temperature as well as pressure. The change in viscosity caused due
to temperature is very crucial in a majority of the practical applications where the lubricants
are expected to perform under different values of temperature (Freeman, 1962). Tipei (1962)
performed an experimental study which suggested that the viscosity-temperature relationship
is substitutable by a establishing a relationship between the viscosity and the film thickness.
The study also suggested that the least film thickness is associated with the highest
temperature. The study by Sinha et al. (1981) focused on lubrication of an extremely small and
an infinitely long short journal bearing. The results proved that changes in viscosity reduce the
load and friction coefficient. In this modern age, various research and articles is in existence
which can help a researchers as well as in the field of research for different kind of bearing
Analysis
75
using Shliomis model such as, (Lin et al., 2013a) in long journal bearing, (Lin et al., 2013b) in
short journal bearing considering non-Newtonian fluid, (Huang & Wang, 2016) in short
bearing while using different forms of viscosity, (Lin, 2016b) in short journal bearing
considering longitudinal roughness and various forms of viscosity. The impact of thermal
effect on Journal bearing is studied by a lot of researchers using a number of different
parameters. For example, (Reddy et al., 2012) used couple stress fluid, (Kumar et al., 2013)
studied the model using two layer fluid considering cavitations, (Siddangouda et al., 2013)
studied roughness, (Naduvinamani et al., 2014) used Micro-polar fluid and different forms of
transverse and longitudinal roughness, (Patel et al., 2018) used Neuringer-Rosensweig model
taking into account smooth roughness.
By reviewing literature concerning Ferrofluid flow, one can understand that Shliomis model
displays better results than Neuringer-Rosensweig model. Thus, this study is focused on
scrutinizing the performance of thermal effect on Ferrofluid lubrication using short bearing
based on Shliomis model.
6.2 Analysis
The Fig. 6.1 shows the geometrical design of the system and its configurations. U denotes the
uniform velocity of the system in the direction x .
FIGURE 6.1 Configuration of the bearing system (Patel & Deheri, 2013a)
The thickness h is,
m sh h h (6.1)
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
76
where mh is taken as (Patel & Deheri, 2013a):
1 22
2
1 1 ,m
h hxh h m m
L h
using the works of (Christensen & Tonder, 1969a,b, 1970). Also, the study uses sh using the
probability density function
32
2
351 ,
32
0 , elsewhere
ss
s
hc h c
f h c c
(6.2)
In this case, c is the highest possible deviation from the average width of the film. , and
are considered in view of relationship
2 2 3( ), ( ) , ( )s s sE h E h E h (6.3)
where ( )E denotes the expectancy operator known by
( ) ( ) ( )
c
s s
c
E f h dh
(6.4)
Actually, magnetic fluids or the Ferrofluids have a constant nature and are the type of
colloidal suspensions that possess extremely superior magnetic particles in a viscous fluid. We
can use an external magnetic field to position, limit or monitor these fluids as required. This,
in turn, increases the fluid’s effective viscosity. This research has substantially contributed to
the increase application of these magnetic fluids in bearing systems as lubricating agents. An
important point here is that most of the studies based on Neuringer-Rosensweig model have
concluded that the vector of magnetization is parallel to the applied magnetic field. Since
Shliomis model considers particle rotation, it overcomes this limitation.
Shliomis (1974) proposed that a change in applied magnetic field can have a two-way
implication on the particles in a magnetic fluid. Either, the rotation of such particles is
impacted or the magnetic moment in them changes. B (Brownian relaxation time parameter)
is used to derive the particle rotation and S (relaxation time parameter) suggests the intrinsic
Analysis
77
process of rotation. Considering a steady flow while overlooking the second derivatives and
inertial of S , the revised flow equation becomes,
2
0
1. 0
2 s
p I
q M H S (6.5)
1
2 q (6.6)
0 sI S M H (6.7)
0BM
H I
HM S M (6.8)
0 H (6.9)
and
0 H M (6.10)
The following can be derived by combining all the above mentioned equations
2
0 0
1. 0
2p q M H M H
(6.11)
00
B sBM
H I
HM M M H M (6.12)
Langevin’s parameter 1 is used for the strong magnetic field, the above equation changes
to
0B
M
H M H H (6.13)
with
6
1 cothBnk T
(6.14)
where
0
0
1coth , Bk T
M n H
(6.15)
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
78
For a deferment of spherical particles:
6s
I
and
3B
B
V
k T
(6.16)
With uniform magnetic field 00, ,0HH , (6.11) to (6.13) develop into the following
(Majumdar, 2008)
1 0 1p u p p w
x y y y z y y
(6.17)
and
( ) ( ) ( ) 0u v wt x y z
(6.18)
where (referring to Shliomis, 1974)
3 tanh
,2 tanh
(6.19)
and
0
2
q
h
h
(6.20)
The above equation denotes thermal variation considering the viscosity-temperature, when the
viscosity 0 at 2mh h , whereas q , the thermal factor, usually maintain the value between 0
and 1 according to the nature of the lubrication (Tipei, 1962).
Solving (6.17) under no slip boundary conditions (Majumdar, 2008);
at 0, ,b by u u w w and
at , ,a ay h u u w w
One can find
1( )
2 (1 )
1( )
2 (1 )
b a
b a
p h y yu y y h u u
x h h
p h y yw y y h w w
z h h
(6.21)
Analysis
79
Putting the velocity components expression into continuity equation (6.18) and integrating
under the conditions bv v at 0y and av v at y h gives rise to;
0 0
0 0
1( ) ( )
2 (1 ) (1 )
h h
a b
h h
b a b a
p pv v y y h dy y y h dy
x x z z
h y y h y yu u dy w w dy
x h h z h h
ht
(6.22)
In reference to experiment through the utilization of relation (Majumdar, 2008)
0 0
( , , ) ( , , ) ( , , )h
f x y z dz f x y z dz f x y hx x x
h h
(6.23)
one obtains for constant velocities
3 3
12 (1 ) 12 (1 ) 2
h p h p Uh h
x x z z x t
(6.24)
where
a bU u u (6.25)
The generalized Reynolds’ equation (6.24) turns in the state of equilibrium, which brings
(Majumdar, 2008)
3 3
12 (1 ) 12 (1 ) 2
h p h p Uh
x x z z x
(6.26)
Modified two dimensional Reynolds’ equation, for the incompressible lubricant is
3 3
12 (1 ) 12 (1 ) 2
h p h p U h
x x z z x
(6.27)
In the x -direction, the flow due to pressure gradient in the variation of pressure can be
avoided when the bearing is short. In this case, the one dimensional equation (6.27) lead to:
3 6 1p h
h Uz z x
(6.28)
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
80
which is the Reynolds’ equation (Basu et al., 2005; Majumdar, 2008; Shimpi & Deheri, 2010)
modified according to the general hydrodynamic lubrication assumptions.
According to the stochastically average process of (Shimpi & Deheri, 2010), (6.28) becomes:
3 6 1p
E h U E hz z x
, , , 6 1m m
pg h U h
z z x
(6.29)
where
3 2 2 2 2 3, , , 3 3 3m m m mg h h h h (6.30)
The ensuing dimensionless quantities are used
3
2 2
3
2
3 2
2 2 2 0 2 2
, , ,, , 1 1 , , , ,
, , , , ,
mmg hhx z
X Z h m X g hL B h h
ph L Bp L B
h h h UB h h
(6.31)
The related pressure limit settings are (Deheri & Patel, 2011)
0p at 1
2Z and
d0
d
p
Z at 0Z (6.32)
With the aid of (6.32) the pressure distribution in dimensionless form overcome with
2
3 1 1 1 1
4, , ,
q
m m Xp Z
L g h
(6.33)
where
3 2 2 2 2 3
, , , 3 3 3g h h h h (6.34)
The bearing ability of dimensionless type (Patel & Deheri, 2013a) is obtained
3
2
4
0
h WW
UB (6.35)
Analysis
81
1
0
1 11d
2 , , ,
q
m XmW X
B g h
(6.36)
where the load bearing capacity is calculated using
2
0
2
, d d
B
L
B
W p x z x z
(6.37)
6.3 Results and Discussion
Equation (6.33) shows that the pattern of pressure distribution when in a dimensionless form.
Additionally, the bearing system’s capacity can be derived using (6.36) in a non-dimensional
form. When , the parameter of roughness is assumed to be 0, this investigation diminishes to
the study of a Ferrofluid based short bearing. If the magnetization constant is also considered
0, the study becomes a performance analysis of the system as suggested by (Basu et al., 2005).
1/3 rule as given by Simpson having a step size of 0.2 is helpful in calculating (6.36) for
altering the measure of magnetization parameter , thermal factor q , aspect ratio m ,
roughness parameters , , .
The final values of W using different parameters are plotted graphically. The Figs. 6.2(a to f)
display the changes in W corresponding to different values of . They suggest that
magnetization leads to a substantial increase in W . It is noticed as of Fig. 6.2(c) that the
amount of minimum W derived is 0.0183 at higher value of with regards to . From Figs.
6.3(a to e) it can be suggested that with an increase in W , q shifts from 0 to 1. Figs. 6.4(a to
d) display the impact of aspect ratio on W and suggests that aspect ratio causes a sharp
increase in W . The decrease in the value of W in Figs. 6.5(a to c) displays the unfavorable
impact of standard deviation on a bearing system’s performance. Figs. 6.6(a to b) establish an
inverse relationship between variance and W suggesting that positive variance reduces W
while negative variance increases it. Under the impact of skewness, W changes according to
Fig. 6.7.
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
82
(a) q
(b) m
(c)
Results and Discussion
83
(d)
(e)
(f) B
FIGURE 6.2 Profile of W with regards to
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
84
(a) m
(b)
(c)
Results and Discussion
85
(d)
(e) B
FIGURE 6.3 Profile of W with regards to q
(a)
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
86
(b)
(c)
(d) B
FIGURE 6.4 Profile of W with regards to m
Results and Discussion
87
(a)
(b)
(c) B
FIGURE 6.5 Profile of W with regards to
Lubrication of rough short bearing on Shliomis model by ferrofluid considering viscosity
variation effect
88
(a)
(b) B
FIGURE 6.6 Profile of W with regards to
FIGURE 6.7 Profile of W with regards to
Conclusions
89
6.4 Conclusions
The Shliomis model of Ferrofluid and the stochastic theory by Christensen have been used as
the basis of this study to analyze the impact of changes in Ferrofluid lubrication viscosity in
the case of short bearings. The pressure distribution and the bearing system’s capacity were
analyzed numerically. The results obtained lead to the following conclusion:
• Thermal effect has an unfavorable impact on a system’s load bearing capacity.
• Magnetic strength in appropriate measures can be used to nullify the impact of the
thermal effect.
• A left-ward skewed surface roughness increases the bearing capacity and a right-ward
skewed surface roughness decreases the bearing capacity.
• This study is insightful since it ensures a higher freedom of magnitude than (Verma,
1986) and (Prajapati, 1994).
• The decrease in the bearing system’s capacity due to roughness standard deviation can
be nullified by .
• When used magnetic fields, Ferrofluid increase the capacity of short bearings in
contrast to the systems functioning with conventional bearing.
This study provides new insight on improving bearing system’s capacity. Additionally, the
findings of this study support the theory stating that a bearing system’s life can be starkly
improved by controlling the loss of lubricants.
90
CHAPTER 7
A Study of Ferrofluid Lubrication Based Rough Sine
Film Slider Bearing with Assorted Porous Structure
7.1 Introduction
The last decade has seen a considerable shift wherein many tribological researches have been
dedicated to study surface roughness and the impact of hydrodynamic lubrication. This is
because every solid surface carries some amount of surface roughness, the height of which is
usually parallel to the mean separation between lubricated contacts. As many researchers have
suggested, studying the surface roughness will help to improve the performance of a bearing
system. Due to this reason, many researchers (Andharia et al., 2001; Naduvinamani & Biradar,
2007; Naduvinamani et al., 2015) studied the performance of various bearing systems using
the stochastic concept of (Christensen & Tonder, 1969a,b, 1970).
Amongst the biggest inventions in the field is the use of Ferrofluid as a bearing system
lubricant. A number of authors (Bhat, 2003; Hamrock, 1994; Neuringer & Rosensweig, 1964;
Patel et al., 2017b; Vashi et al., 2018) have worked to explain the performance and
applications of Ferrofluid when used in different types of bearing systems. These studies have
suggested that Ferrofluid impacts the bearing performance positively.
Many researchers have used different types of film geometries in order to study the effect of
Ferrofluid based squeeze film. Some of the researches conducted on this topic are listed, Shah
and Bhat (2003a) studied exponential slider bearing, Shah and Bhat (2003b) worked on secant
shaped slider bearing, Naduvinamani and Apparao (2010), Patel and Deheri (2012) and Ram
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
91
and Verma (1999) analyzed inclined slider bearing, Singh (2011) studied curved slider
bearings, Patel and Deheri (2016b) examined parallel slider bearing, Shukla and Deheri (2013)
evaluated on Rayleigh step bearing, Patel and Deheri (2014c) investigated parabolic slider
bearing, Patel et al. (2014) worked on hyperbolic slider bearing, Lin (2016a) discussed sine
film thrust bearing, Deheri et al. (2016) studied convex pad slider bearing and Patel and
Deheri (2016a) examined infinitely long slider bearing. From all the articles above, it can
clearly be seen that the characteristics of Ferrofluid and its effect on load bearing capacity are
positive.
The lubrication theory of porous bearings was first studied by (Morgan & Cameron, 1957).
Porous structures are usually described using two common parameters, which are porosity and
permeability. Porosity is a measure of existing voids within a dense material structure.
Permeability defines the ease with which fluids can flow through the material, in case of open
cell porosity. Darcy’s law is generally used to determine the porosity. Porous metallic
materials have a lot of applications including vibration and sound absorption, light materials,
heat transfer media, sandwich core for different panels, various membranes and during the last
years as suitable biomaterial structures for design of medical implants. Porous matrix
decreases the load carrying capacity and increase the frictional force on the slider. The porous
layer has a beneficial property of self-lubrication, making it an important area of study. Patel
and Deheri (2013a) worked on studying the comparisons of porous structures and their impact
on the load carrying capacity of a magnetic fluid based rough and short bearing. The studies
have found that while magnetization has a positive impact on the bearing system’s
performance, transverse roughness impacts it negatively. However, in the case of Kozeny-
Carman model, this negative impact is comparatively lower. In this model, the negative
impact of porosity on the bearing performance can be neutralized with the negatively skewed
roughness’ positive impact. Patel and Deheri (2014a) worked on investigating the
performance of a magnetic fluid based double layered rough porous slider bearing considering
the combined porous structures. For a considerable range of combined porous structure,
magnetization neutralizes the adverse effect of roughness. Patel and Deheri (2014b) studied
Shliomis model-based magnetic squeeze film in rotating rough curved circular plates: making
a contrast of two different porous structures. It was found out that by choosing a proper
rotation ratio and appropriate curvature parameters, the negative impacts of transverse
Introduction
92
roughness on a bearing’s load carrying capacity can be nullified by the positive impact of
magnetization with negatively skewed roughness. Shah and Patel (2012) studied squeeze film
based on Ferrofluid in curved porous circular plates with various porous structures. The
studies showed that, with concave plates and porous structure given by Kozeny-Carman, there
was a considerable increase in the load bearing capacity. Different forms of modification of
Darcy’s law have been studied in (Prajapati, 1995). Barik et al. (2016) investigated a bearing
system based on a hyperbolic slider. They experiment with porous structure as well as
roughness in accordance with the impact of sinusoidal magnetic field. Furthermore, the load
bearing capacity is enhanced due to the influence of magnetization and the slip parameter
being within the limited boundary. Recently, (Mishra et al., 2018) analyzed inclined slider
bearing. In this work, we can identify that they worked in detail with all aspects of surface
roughness, porosity and magnetic field. Somehow, by surprise, the result was that the load
bearing capacity differs and gives a very effective ability when the sinusoidal magnetic field is
applied in the form which appears in the presented study.
None of the above-mentioned researchers worked on the impact of sine films in a slider
bearing. In order to explore this filed, this paper studies Ferrofluid lubrication based rough
sine film slider bearing with assorted porous structure.
7.2 Analysis
The Fig. 7.1 shows the geometrical design of the system and its configurations. U denotes the
uniform velocity of the system in the direction x .
The thickness h is considered as
m sh h h (7.1)
where mh is taken as (Lin, 2016a):
2 1 2 1 sin2
m
xh h h h
L
using the works of (Christensen & Tonder, 1969a,b, 1970).
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
93
FIGURE 7.1 Configuration of a sine film porous slider bearing including squeeze action (Lin, 2016a)
Also, the study uses sh using the probability density function
32
2
351 ,
32
0 , elsewhere
ss
s
hc h c
f h c c
(7.2)
c being the maximum deviation from the mean film thickness. , and are considered
by the relationships
2 2 3( ), ( ) , ( )s s sE h E h E h (7.3)
where ( )E denotes the expectancy operator given by
c
s s
c
E f h dh
(7.4)
Neuringer and Rosensweig (1964) formulated explaining the steady flow of a magnetic fluid.
It was:
Equation of motion
2
0( . ) ( . )p q q q M H (7.5)
Analysis
94
Equation of magnetization
M H (7.6)
Equation of continuity
. 0 q (7.7)
Maxwell equations
0 H (7.8)
and
. 0 H M (7.9)
where , , , p q M and are fluid density, fluid velocity, magnetization vector, film pressure
and fluid viscosity respectively.
Also,
ui vj wk q (7.10)
where , ,u v w are components of film fluid velocity in ,x y and z - directions respectively.
Further, the magnetic field’s magnitude is given by
2 ( )H k x L x (7.11)
where k is a suitable constant and, assuming the external magnetic field to come up from a
potential function, the inclination angle of the magnetic field ( , )x z satisfies the equation
(Bhat, 2003)
2cot
2 ( )
x L
x z x L x
(7.12)
The governing equation of motion of the fluid flow in the film region (Verma, 1986) is
22
02
1 1
2
up H
z x
(7.13)
By solving (7.13) following the no slip boundary conditions:
0u at z h and u U at 0z
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
95
One can find
211
2 2
z h p zu z U
x h
(7.14)
Integrating (7.14) over the film region, yields
3
012 2
hh dp Uh
u dzdx
(7.15)
Using (7.15) in continuity equation
0
0
0
h
hu dz w wx
(7.16)
yields
3
0 012 2
h
h dp Uhw w
x dx
(7.17)
where
0hw h
and 0 0w
Equation (7.17) leads to:
3 2
0 0
16 12
2
d d dhh p H U h
dx dx dx
(7.18)
which is the Reynolds’ equation (Bhat, 2003; Patel et al. 2014) modified according to the
general hydrodynamic lubrication assumption.
According to the stochastically average process of (Christensen & Tonder, 1969a), (7.18)
becomes:
3 2
0 0
16 12
2
d d dE h p H U E h h
dx dx dx
1/32
0 0
1, , , , 6 , , , , 12
2m m
d d dg h K p H U g h K h
dx dx dx
(7.19)
where
Analysis
96
3 2 2 2 2 3
1, , , , 3 3 3 12m m m mg h K h h h Kl
(7.20)
7.2.1 A Globular Sphere Model
Globular particles (a mean particle size Dc) are used to fill a porous material which is given in
Fig. 7.2.
FIGURE 7.2 Structure model of porous sheet given by Kozeny‐Carman (Yazdchi et al., 2011)
In fluid dynamics, the Kozeny‐Carman equation (Carman, 1937) plays a major role in
calculating the pressure drop when working with a fluid flowing in a packed bed of solids.
Although, the equation only remains valid for a laminar flow. This equation makes use of few
general experimental trends, which makes it an efficient quality control tool that can be used
for both physical as well as digital experimental results. The equation is commonly displayed
as permeability versus porosity, pore size and tortuosity.
The pressure gradient is assumed to be linear here. Following the ideas of discussion (Liu,
2009) the use of Kozeny‐Carman formula becomes:
2 3
272(1 )
cD lK
l
where is the porosity and l l is the length ratio.
The following dimensionless quantities are used,
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
97
1
3
2 2 2
1/3 21/3
2
3
2 2 2 2
2 2 3 2
0 2 121 2 3
20
, , , ,, , 1 ( 1) 1 sin , , , , , ,
2
, , , ,, , , , , , , , ,
, , , ,72(1 )2
mm
m
c c
g h KhhxX a h a X g h K
L h h h
g h K phg h K p
h h h h UL
h Lk D l D lUh ll K K
U l l hh L
(7.21)
The associated boundary conditions are
0p at 0,1X (7.22)
With the aid of (7.22) the pressure distribution in a non-dimensional form comes out to be
1/3 1/31
1
0
, , , , 1, , , , (1 )1(1 ) 6
2 , , , ,
X g h K g K Xp X X dX
g h K
(7.23)
where
3
3 2 2 2 2 3
3, , , , 3 3 3
6(1 )
K lg h K h h h
(7.24)
The load bearing capacity in dimensionless form is obtained
2
2
2
hW W
UL B (7.25)
1/3 1/311
1
0
, , , , 1, , , , (1 )6 (1 )
12 , , , ,
g h K g K XW X dX
g h K
(7.26)
where the load bearing capacity is calculated using
0
L
W pB dX (7.27)
Results and Discussion
98
7.3 Results and Discussion
The results calculated for the dimensionless load-carrying capacity W given by (7.26) are
found using Simpson’s one-third rule with a step size 0.2 for the Kozeny-Carman model. It
proves that the load bearing capacity increases by:
12
Equation (7.26) suggests that even in the absence of flow, a bearing system can handle a given
amount of load for the Kozeny-Carman model. By keeping the roughness zero, the study
reduces to the impact of an assorted porous structure on the Neuringer-Rosensweig model
based Ferrofluid squeeze film for a slider bearing (Bhat, 2003). Considering the magnetization
parameter as a zero, it reduces to the study of (Basu et al., 2009) in the absence of porosity.
Equation (7.26) clearly suggests that the expression for W is linear with respect to the
magnetization parameter . Thus when the Kozeny-Carman model is applicable, by
increasing magnetization, the load bearing capacity can also be increased (Fig. 7.3). Figs. 7.3-
7.10 display a graphical representation of the Kozeny-Carman model results. They suggest
that:
1) According to Fig. 7.4, it is evident that a standard deviation has a relatively lower
impact when compared to porosity.
2) As the positive variance increases, the load carrying capacity decreases. A decrease in
the negative variance leads to an increase in the load carrying capacity (Fig. 7.5). As
suggested by Fig. 7.6, the impact of skewness on the load carrying capacity is similar
to variance.
3) Effect of K on W with respect to and l is seen to be adversely from Fig. 7.7.
4) Fig. 7.8 demonstrates the impact of porosity on the distribution of load carrying
capacity. It suggests that porosity considerably reduces the load bearing capacity. In
case of a measure of symmetry, this scenario is further exaggerated.
5) Fig. 7.9 displays the impact of the ratio l on the load bearing capacity. It is evident
that, with an increase in l , the load bearing capacity decreases. The rate of it is further
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
99
increased with an increase in porosity parameter . With the further investigation the
investigator went through physical point of view that Maximum load-carrying capacity
can be obtained when maximum magnetic field and center of pressure coincide.
Moreover, porous surface is inserted because of advantageous property of self-
lubrication. It should be noted here that, when porous layer is inserted then the
pressure of the porous medium provides a path for the fluid to come out easily from
the bearing to the environment, which varies with permeability. Thus, the presence of
the porous material decreases the resistance to flow in r-direction and as a consequence
the load carrying capacity decreases with increasing values of porosity parameter.
If we look on the results presented in Fig. 7.10a and correlate them with the results from Fig.
10, we can firmly conclude that the Kozeny-Carman model is highly activated in reference to
the conventional porosity case.
(a)
(b)
0.1640
0.1651
0.1662
0.1673
0.1684
0.01 0.02 0.03 0.04 0.05
W̅
µ*
σ̅ = 0.01
σ̅ = 0.03
σ̅ = 0.05
σ̅ = 0.07
σ̅ = 0.09
0.1620
0.1650
0.1680
0.1710
0.1740
0.01 0.02 0.03 0.04 0.05
W̅
µ*
α̅ = -0.02
α̅ = -0.01
α̅ = 0
α̅ = 0.01
α̅ = 0.02
Results and Discussion
100
(c)
(d) K
FIGURE 7.3 Profile of W with regards to
(a)
0.1480
0.1495
0.1510
0.1525
0.1540
0.01 0.02 0.03 0.04 0.05
W̅
µ*
ε̅ = -0.02
ε̅ = -0.01
ε̅ = 0
ε̅ = 0.01
ε̅ = 0.02
0.1650
0.1662
0.1674
0.1686
0.1698
0.01 0.02 0.03 0.04 0.05
W̅
µ*
K̅ = 10
K̅ = 20
K̅ = 30
K̅ = 40
K̅ = 50
0.1680
0.1705
0.1730
0.1755
0.1780
0.01 0.03 0.05 0.07 0.09
W̅
σ̅
α̅ = -0.02
α̅ = -0.01
α̅ = 0
α̅ = 0.01
α̅ = 0.02
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
101
(b)
(c) K
(d) l
FIGURE 7.4 Profile of W with regards to
0.1540
0.1547
0.1554
0.1561
0.1568
0.01 0.03 0.05 0.07 0.09
W̅
σ̅
ε̅ = -0.02
ε̅ = -0.01
ε̅ = 0
ε̅ = 0.01
ε̅ = 0.02
0.1690
0.1695
0.1700
0.1705
0.1710
0.01 0.03 0.05 0.07 0.09
W̅
σ̅
K̅ = 10
K̅ = 20
K̅ = 30
K̅ = 40
K̅ = 50
0.1696
0.1700
0.1704
0.1708
0.1712
0.01 0.03 0.05 0.07 0.09
W̅
σ̅
l*= 1.75
l*= 1.95
l*= 2.15
l*= 2.35
l*= 2.55
Results and Discussion
102
(a)
(b) K
FIGURE 7.5 Profile of W with regards to
FIGURE 7.6 Profile of W with regards to
0.1613
0.1633
0.1653
0.1673
0.1693
-0.02 -0.01 0 0.01 0.02
W̅
α̅
ε̅ = -0.02
ε̅ = -0.01
ε̅ = 0
ε̅ = 0.01
ε̅ = 0.02
0.1677
0.1697
0.1717
0.1737
0.1757
-0.02 -0.01 0 0.01 0.02
W̅
α̅
K̅ = 10
K̅ = 20
K̅ = 30
K̅ = 40
K̅ = 50
0.1456
0.1460
0.1464
0.1468
0.1472
-0.02 -0.01 0 0.01 0.02
W̅
ε̅
K̅ = 10
K̅ = 20
K̅ = 30
K̅ = 40
K̅ = 50
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
103
(a)
(b) l
FIGURE 7.7 Profile of W with regards to K
FIGURE 7.8 Profile of W with regards to
0.1230
0.1329
0.1428
0.1527
0.1626
10 20 30 40 50
W̅
K̅
ψ = 0.15
ψ = 0.2
ψ = 0.25
ψ = 0.3
ψ = 0.35
0.1590
0.1602
0.1614
0.1626
0.1638
10 20 30 40 50
W̅
K̅
l*= 1.75
l*= 1.95
l*= 2.15
l*= 2.35
l*= 2.55
0.1404
0.1454
0.1504
0.1554
0.1604
0.1 0.15 0.2 0.25 0.3
W̅
ψ
l*= 1.75
l*= 1.95
l*= 2.15
l*= 2.35
l*= 2.55
Results and Discussion
104
(a)
(b)
FIGURE 7.9 Profile of W with regards to l
(a)
0.1750
0.1765
0.1780
0.1795
0.1810
1.75 1.95 2.15 2.35 2.55
W̅
l*
µ* = 0.01
µ* = 0.02
µ* = 0.03
µ* = 0.04
µ* = 0.05
0.1260
0.1340
0.1420
0.1500
0.1580
1.75 1.95 2.15 2.35 2.55
W̅
l*
ψ = 0.15
ψ = 0.2
ψ = 0.25
ψ = 0.3
ψ = 0.35
0.1360
0.1450
0.1540
0.1630
0.1720
0.01 0.02 0.03 0.04 0.05
W̅
µ*
ψ = 0.15
ψ = 0.2
ψ = 0.25
ψ = 0.3
ψ = 0.35
ψ*= 0.01
ψ*= 0.02
ψ*= 0.03
ψ*= 0.04
ψ*= 0.05
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
105
(b)
(c)
FIGURE 7.10 Profile of W with regards to , and for the comparison of and
7.4 Validation
Undoubtedly, Tables 7.1-7.5 underline that an enhancement in the load bearing capacity by
almost 5% is registered here.
TABLE 7.1 Comparison of W calculated for
Quantity Load carrying Capacity (calculated for
10.01, 0.01, 0.05, 30,1 0.01, 1.75, 0.15, 0.02K l )
Result for assorted porosity Result for conventional porosity
0.01 0.1651655 0.1567618
0.02 0.1659988 0.1575952
0.03 0.1668321 0.1584285
0.1330
0.1390
0.1450
0.1510
0.1570
0.01 0.03 0.05 0.07 0.09
W̅
σ̅
ψ = 0.15
ψ = 0.2
ψ = 0.25
ψ = 0.3
ψ = 0.35
ψ*= 0.01
ψ*= 0.02
ψ*= 0.03
ψ*= 0.04
ψ*= 0.05
0.1300
0.1370
0.1440
0.1510
0.1580
-0.02 -0.01 0 0.01 0.02
W̅
ε̅
ψ = 0.15
ψ = 0.2
ψ = 0.25
ψ = 0.3
ψ = 0.35
ψ*= 0.01
ψ*= 0.02
ψ*= 0.03
ψ*= 0.04
ψ*= 0.05
Validation
106
0.04 0.1676655 0.1592618
0.05 0.1684988 0.1600952
TABLE 7.2 Comparison of W calculated for
Quantity Load carrying Capacity (calculated for
10.02, 0.01, 0.05, 30,1 0.01, 1.75, 0.15, 0.02K l )
Result for assorted porosity Result for conventional porosity
-0.02 0.1716939 0.1623983
-0.01 0.1697629 0.1607764
0.00 0.1678648 0.1591755
0.01 0.1659988 0.1575952
0.02 0.1641643 0.1560354
TABLE 7.3 Comparison of W calculated for
Quantity Load carrying Capacity (calculated for
10.02, 0.01, 0.05, 30,1 0.01, 1.75, 0.15, 0.02K l )
Result for assorted porosity Result for conventional porosity
0.01 0.1659988 0.1575952
0.03 0.1658797 0.1574952
0.05 0.1656424 0.1572959
0.07 0.1652885 0.1569985
0.09 0.1648207 0.1566048
TABLE 7.4 Comparison of W calculated for
Quantity Load carrying Capacity (calculated for
10.02, 0.01, 0.03, 30,1 0.01, 1.75, 0.15, 0.02K l )
Result for assorted porosity Result for conventional porosity
-0.02 0.1692074 0.1602794
-0.01 0.1687180 0.1598716
0.00 0.1682335 0.1594672
0.01 0.1677536 0.1590663
0.02 0.1672784 0.1586686
TABLE 7.5 Comparison of W calculated for
Quantity Load carrying Capacity (calculated for
10.02, 0.01, 0.01, 30, 0.01,1 0.01, 0.8, 0.02K l )
Result for assorted porosity Result for conventional porosity
0.10 0.1706303 0.1599767
0.15 0.1699394 0.1599767
0.20 0.1684007 0.1599767
0.25 0.1655293 0.1599767
0.30 0.1607807 0.1599767
7.5 Conclusions
This paper has studied the effect of Ferrofluid lubrication when used with a rough sine film
slider bearing with an assorted porous structure on the load carrying capacity. A modified
A study of ferrofluid lubrication based rough sine film slider bearing with assorted porous
structure
107
Reynolds’ equation used for the sine profile slider bearing lubrication has been derived with
the ferrohydrodynamic theory by Neuringer-Rosensweig and equation of continuity for film as
well as porous region. The Reynolds’ equation has also been used to determine the pressure
equation and an expression for dimensionless load-carrying capacity. From the numerical
calculations, the following conclusions have been derived:
• By increasing the strength of the external magnetic field, a bearing system’s pressure
and its load bearing capacity can be increased considerably. Also, unlike conventional
lubricants, this type of a system can carry a given amount of load even if there is no
flow. Additionally, as suggested by (7.13), when the Neuringer-Rosensweig Ferrofluid
flow model is applicable, a constant magnetic field does not increase the load bearing
capacity.
• Comparing the present paper with (Patel & Deheri, 2012) makes it evident that the
system, in this case, enhances the load carrying capacity threefold at minimum. Also,
when a sine film profile is used to design the slider bearing, it enhances the bearing
capacity, as can be seen when compared with an inclined slider bearing.
Lastly, the article determines that, when Kozeny-Carman’s model is appropriate, the surface
roughness must be studied properly in order to design a more efficient bearing system.
108
CHAPTER 8
General Conclusion and Future Scope
General Conclusion
The study explores and identifies that though transverse surface roughness has a negative
impact on the load bearing capacity in general, the performance can be improved by using
negatively skewed roughness along with negative variance. In a further research it is also
found that, the attempts made to neutralize the adverse impacts of surface roughness; porosity
and slip velocity with the help of magnetization are considerably limited. However, the
negative impact of roughness displays more variation in this situation despite the fact that
standard deviation raises the load bearing capacity.
This research tries to carry out new concept by performing and applying the various theories
as well as pattern and model to explore the new concept of research horizon. The focus is to
develop the capacity of bearing system by the help of stochastic averaging roughness model
which was introduced by Christensen and Tonder. The researcher also sets the aim to get
positive effect and so, they utilized the concept of magnetic fluid flow models of Shliomis and
Neuringer-Rosensweig. The assorted porous structure by Kozeny-Carman model was also
experimented in the study.
Interrogation of study try not to limit its’ boundary however also performs with the surface
roughness should be a primary concern with the designs of magnetic fluid based bearing
system. The report also suggests that for a ‘no flow’ situation, the bearing can endure only a
specific load amount.
General conclusion and future scope
109
Shliomis’ Ferrofluid flow provides relevant insights on the impact of rotations of the career
liquid and magnetic particles. Further, a varying magnetic field provides the benefit of
creating the maximum field according to the necessary contact area of the bearing. Further,
this study can create a new pathway for ensuring maximum utilization of a bearing system. It
also clearly proposes that by managing the lubricant loss, the life span of a load bearing
system can be increased substantially.
The study also focuses on thermal effect and its vital role as well because it represents the
nonmetallic effect. The Shliomis model of Ferrofluid and the stochastic theory by Christensen
have been used as the basis of this study to analyze the impact of changes in Ferrofluid
lubrication viscosity in the case of short bearings. Thermal effect has a negative impact on a
system’s load bearing capacity. Magnetic strength in appropriate measures can be used to
nullify the impact of the thermal effect.
The study related to effects of slip velocity is calculated by using the slip model of Beavers
and Joseph not only that, the model of Morgan and Cameron introduced hydrodynamic
lubrication theory of bearings with porous structure is also included in research, In addition
Tipei model represented Viscosity Variation Effect in same manner. Concluded with, if the
accuracy and appropriacy works hand to hand the result can be found in a positive manner.
The researcher believes that research is not a profession but it is a passion and so he tries to
deal with his best.
Future Scope
This study aims to perform a comprehensive analysis of the following:
• The impact caused by deformation in the load bearing capacity of different bearing
systems can be studied.
• The models of magnetic fluid flow (Jenkins, 1972; Neuringer & Rosensweig, 1964;
Shliomis, 1974) can be compared so as to know in which particular model load bearing
capacity is in high proportion.
• We may focus on applying double layered porous structure to the various bearings.
• Possibilities for the application of hydromagnetic lubrication to the bearings to
improve their load carrying capacity can be examined.
Future scope
110
• It is also possible to study the theoretical implications concerning the impact of a
system’s roughness on the type and features of lubrication used with the help of
micropolar fluid.
• Ample of scope to front forward with profile of the piston top compression ring face
which is assumed to be a parabola is also found.
• The Jenkins model of fluid flow may be used in order to study the ways in which
deformation can impact different types of bearing systems.
• The impact caused by couple stress can be studied with the help of magnetic fluid
flow.
• To explore the research on annular plates with all the parameters which are utilized in
the study.
• Analysis of the surface topology of the bearing system.
By focusing on such a diverse range of topics, this study becomes relevant to various different
streams of engineering and science including physics, material science, mechanical
engineering, mathematics, etc.
111
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List of Publications
1. Effect of Slip Velocity on a Magnetic Fluid Based Squeeze Film in Rotating
Transversely Rough Curved Porous Circular Plates. Industrial Engineering Letters,
7(8), 28-42, 2017.
http://www.iiste.org/Journals/index.php/IEL/article/view/40253/41400 (UGC listed)
2. Analysis of Rough Porous Inclined Slider Bearing Lubricated With a Ferrofluid
Considering Slip Velocity. International Journal of Research in Advent Technology,
7(1), 387-396, 2019. doi.org/10.32622/ijrat.71201977 (UGC listed)
3. A Study of Ferrofluid Lubrication Based Rough Sine Film Slider Bearing With
Assorted Porous Structure. Acta Polytechnica, 59(2), 144-152, 2019.
doi.org/10.14311/ap.2019.59.0144 (Web of Science and Scopus indexed, UGC listed)
4. Lubrication of Rough Short Bearing on Shliomis Model by Ferrofluid Considering
Viscosity Variation Effect. International Journal of Mathematical, Engineering and
Management Sciences, 4(4), 982-997, 2019. doi.org/10.33889/IJMEMS.2019.4.4-078
(Web of Science and Scopus indexed, UGC listed)
5. Influence of Ferrofluid Lubrication on Longitudinally Rough Truncated Conical Plates
with Slip Velocity. Mathematical Journal of Interdisciplinary Sciences, 7(2), 93-101,
2019. doi.org/1015415/mjis.2019.72012 (UGC listed)
6. Numerical Modelling of Shliomis Model Based Ferrofluid Lubrication Performance in
Rough Short Bearing. Journal of Theoretical and Applied Mechanics, 57(4), 923-934,
2019. doi.org/10.15632/jtam-pl/112415 (Web of Science and Scopus indexed, UGC
listed)
7. Effect of Slip Velocity on a Ferrofluid based Longitudinally Rough Porous Plane
Slider Bearing. In: K. N. Das et al. (eds) Proceeding of 8th International conference on
Soft Computing for Problem Solving-SocProS 2018, VIT-Vellore, Tamil Nadu, India,
17-19 December 2018. Singapore: Advances in Intelligent Systems and Computing
series of Springer, 1048, 27-41, 2020. doi.org/10.1007/978-981-15-0035-0_3 (Scopus
indexed, UGC listed)
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