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

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

iv

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

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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:


vi

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Place: Kalol



Place: Ahmedabad

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Place: Kalol



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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.

xii

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.

xiv

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


4.4 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


6.4 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


7.4 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



3.7 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








5.4 Profile of W with regards to m ……………………………………………………………... 68






6.3 Profile of W with regards to q ……………………………………………………………… 85

6.4 Profile of W with regards to m ……………………………………………………………... 86


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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



7.6 Profile of W with regards to …………………………………………………………….... 102

7.7 Profile of W with regards to K ……………………………………………………………... 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





3.6 Comparison of W calculated for K ……………………………………………………………. 40




5.4 Comparison of W calculated for m ……………………………………………………………. 71

5.5 Comparison of W calculated for B ……………………………………………………………. 71






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


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)


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)


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)


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)


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.


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


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)


( 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)


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)


( ) 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.


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)


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


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)


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)


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)


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)


velocity

34

(f)

FIGURE 3.2 Profile of W with regards to s

(a)

(b)


35

(c)

(d)

FIGURE 3.3 Profile of W with regards to K

(a)


velocity

36

(b)

(c)

(d) K

FIGURE 3.4 Profile of W with regards to


37

(a)

(b)


(a)


velocity

38

(b)



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.


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





-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





0.1 0.2187241 0.2088460

0.2 0.2333267 0.2223130

0.3 0.2576643 0.2447581


velocity

40

0.4 0.2917369 0.2761812

0.5 0.3355446 0.3165824





-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



(calculated for 0.01, 0.05, 0.3, 0.05, 0.7, 0.03K s )


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


(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.


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:


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)


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)



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)


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


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)


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


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.


49

(a)

(b)

(c)


50

(d)

(e) s


(a)


51

(b)

(c)

(d) s



52

(a)

(b)

(c) s



53

(a)

(b) s



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.


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,


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)


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);


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


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


(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)


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


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)


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


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


65

(e)

(f) B


(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


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


67

(e) B


(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


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


69

(b)

(c) B


(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


short bearing

70

(b) B



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%.


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



0.09, 0.02, 0.05, 0.1, 0.1, 0.3, 10, 0.5m B L )


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



0.09, 0.02, 0.1, 0.1, 0.1, 0.3, 10, 0.5m B L )


-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


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


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


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.


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 .


The thickness h is,

m sh h h (6.1)


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)


variation effect

78


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


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


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)


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)


2

0

2

, d d

B

L

B

W p x z x z

(6.37)


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.


variation effect

82

(a) q

(b) m

(c)


83

(d)

(e)

(f) B



variation effect

84

(a) m

(b)

(c)


85

(d)

(e) B

FIGURE 6.3 Profile of W with regards to q

(a)


variation effect

86

(b)

(c)

(d) B

FIGURE 6.4 Profile of W with regards to m


87

(a)

(b)

(c) B



variation effect

88

(a)

(b) B



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).


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


M H (7.6)

Equation of continuity

. 0 q (7.7)

Maxwell equations

0 H (7.8)

and

. 0 H M (7.9)



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


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.



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)


0

L

W pB dX (7.27)


98


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


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


100

(c)

(d) K


(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


structure

101

(b)

(c) K

(d) l


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


102

(a)

(b) K



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


structure

103

(a)

(b) l

FIGURE 7.7 Profile of W with regards to K


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


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


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.


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



10.02, 0.01, 0.05, 30,1 0.01, 1.75, 0.15, 0.02K l )


-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



10.02, 0.01, 0.05, 30,1 0.01, 1.75, 0.15, 0.02K l )


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



10.02, 0.01, 0.03, 30,1 0.01, 1.75, 0.15, 0.02K l )


-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



10.02, 0.01, 0.01, 30, 0.01,1 0.01, 0.8, 0.02K l )


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


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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