STUDY OF EFFECT OF VARIOUS TYPES OF BEARING ON LOAD …€¦ · iv CERTIFICATE I certify that the work incorporated in the thesis “Study of effect of various types of bearing on

STUDY OF EFFECT OF VARIOUS TYPES OF

BEARING ON LOAD CAPACITY

A Thesis submitted to Gujarat Technological University

for the Award of

Doctor of Philosophy

in

Science -Maths

by

Yoginibahen Devendrasinh Vashi

Enrolment No: 149997673017

under supervision of

Dr. Rakesh M. Patel

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

February – 2020

STUDY OF EFFECT OF VARIOUS TYPES OF

BEARING ON LOAD CAPACITY

A Thesis submitted to Gujarat Technological University

for the Award of

Doctor of Philosophy

in

Science -Maths

by

Yoginibahen Devendrasinh Vashi

Enrolment No: 149997673017

under supervision of

Dr. Rakesh M. Patel


AHMEDABAD

February – 2020

ii

© Yoginibahen Devendrasinh Vashi

iii

DECLARATION

I declare that the thesis entitled “Study of effect of various types of bearing on load

capacity” 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 September

2019 under the supervision of Prof. Dr. Rakesh Patel , Assistant Professor & Head,

Department of Mathematics, Gujrat Arts and Science college, Ellisebrige

Ahmedabad 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: 24th February 2020

Name of Research Scholar: Yoginibahen Devendrasinh Vashi

Place: Ahmedabad

iv

CERTIFICATE

I certify that the work incorporated in the thesis “Study of effect of various

types of bearing on load capacity” Submitted by Smt. Yoginibahen

Devendrasinh Vashi 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: 24th February 2020

Name of Supervisor: Dr. Rakesh M. Patel

Place: Ahmedabad

v

Course-work Completion Certificate

This is to certify that Mrs. Yoginibahen Devendrasinh Vashi, enrolment no.

149997673017 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/She has been exempted from the course-work (successfully completed

during M.Phil Course)

He/She has been exempted from Research Methodology Course only

(successfully completed during M.Phil Course)

He/She has successfully completed the PhD course work for the partial

requirement for the award of PhD Degree. His/ Her performance in the

course work is as follows-

Grade Obtained in Research

Methodology

(PH001)

Grade Obtained in Self Study Course

(Core Subject)

(PH002)

BC BB

Supervisor’s Sign:

Name of Supervisor: Dr. Rakesh M Patel

vi

Originality Report Certificate

It is certified that PhD Thesis titled “Study of effect of various types of

bearing on load capacity” by Yoginibahen Devendrasinh Vashi has been

examined by us. We undertake the following:

a. Thesis has significant new work / knowledge as compared already published

or are under consideration to be published elsewhere. No sentence, equation,

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e. The thesis has been checked using Turnitin (copy of originality report attached)

and found within limits as per GTU Plagiarism Policy and instructions issued

from time to time (i.e. permitted similarity index <10%).

Signature of the Research Scholar: Date: 24th February 2020


Place: Ahmedabad

Signature of Supervisor: Date: 24th February 2020


Place: Ahmedabad

vii

viii

PhD THESIS Non-Exclusive License to


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Devendrasinh Vashi having Enrolment No.149997673017 hereby grant a non-

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protection.

h) I retain copyright ownership and moral rights in my thesis, and may deal with

the copyright in my thesis, in any way consistent with rights granted by me

to my University in this non-exclusive license.

ix

i) I further promise to inform any person to whom I may hereafter assign or

license my copyright in my thesis of the rights granted by me to my

University in this non-exclusive license.

j) I am aware of and agree to accept the conditions and regulations of PhD

including all policy matters related to authorship and plagiarism.

Signature of the Research Scholar:


Date: 24th February 2020 Place: Ahmedabad

Signature of Supervisor:


Date: 24th February 2020 Place: Ahmedabad

Seal:

x

Thesis Approval Form

The viva-voce of the PhD Thesis submitted by Smt. Yoginibahen

Devendrasinh Vashi (Enrolment No. 149997673017) entitled “Study of effect

of various types of bearing on load capacity” was conducted on Monday 24th

February 2020 at Gujarat Technological University.

(Please tick any one of the following options)

The performance of the candidate was satisfactory. We recommend that he/she

be awarded the PhD degree.

Any further modifications in research work recommended by the panel after 3

months from the date of first viva-voce upon request of the Supervisor or

request of Independent Research Scholar after which viva-voce can be re-

conducted by the same panel again.

(briefly specify the modifications suggested by the panel)

The performance of the candidate was unsatisfactory. We recommend that

he/she should not be awarded the PhD degree.

(The panel must give justifications for rejecting the research work)

----------------------------------------------------- -------------------------------------------------------

Name and Signature of Supervisor with Seal 1) (External Examiner 1) Name and Signature

----------------------------------------------------- --------------------------------------------------------

2) (External Examiner 2) Name and Signature 3) (External Examiner 3) Name and Signature

xi

ABSTRACT

The present thesis is devoted to study the effect of various types of bearing on load capacity.

In this theoretical study mathematical model has been developed for various types of squeeze

film bearing systems such as circular, parallel stepped, conical, circular parallel stepped.

Ferrofluid is used as a lubricant in these bearings. Tribology is one of the most important

subject dealing with friction, wear, and lubrication. If we could control and reduce main

constituents friction and wear of tribology, automatically it increased the service life of

machine elements. This in turn, saves currency. The identification of tribological problems

and their solutions can increase to significant savings. To reduce friction, lubrication plays

an important role. Selection of suitable lubricant in the machine can extend the machine’s

life period as well.

In recent years, extensive research work has been carried out to study the influence

of ferrofluid lubrication on bearing performance. This field has gained a wide range of

devotion due to its extensive use in technological applications like dynamic sealing, heat

dissipation, damping and medical applications like drug targeting, hyperthermia, and cell

separation. In the recent years, surface roughness and its effects on machine design have

been important features which have been widely studied. Some methods have been

suggested to study the consequence of surface roughness on the bearing performance. Due

to the random structure of the surface roughness, a stochastic model for the study of

hydrodynamic lubrication has been developed by Christensen and Tonder. So, the present

study is purposes to analyze the combined influence of ferrofluid and surface roughness on

various types of porous bearings with couple stress. The generalized Reynolds type equation

is derived to obtain the pressure distribution. With appropriate boundary conditions, the

associated Reynolds type equation is solved to get pressure in the film region which, in turn,

gives the load bearing capacity. Obtained results are presented in graphical forms as well as

tabular forms. Comparision is made between ferrofluid based bearing system and

conventional lubricant based bearing system. Tabular analysis reveals that in the presence

of ferrofluid the system shows better performance compared to the conventional lubricant

case.

xii

Acknowledgment

I express my deep gratitude to my research supervisor Dr. Rakesh M Patel, Head,

Department of Mathematics, Gujarat Arts and Science College, Ahmedabad. He has been a

constant supporter throughout the course of my research work. I feel my deep sense of

gratitude to Dr. G. M. Deheri, Former Associate Professor, S. P. university, Vallabh

Vidyanagar, Anand for his inspirational guidance, encouragement and vital discussions

during the course of my research work. I am also very much thankful to my Doctorate

Progress Committee members Dr. H. C. Patel, Professor, L. D. College of Engineering,

Ahmedabad and Dr. H. R. Kataria, Dean-Faculty of Science, M.S.University of Baroda,

who have reviewed my research work time to time and given the effective suggestion in my

research work.

A special debt of gratitude is owed to Dr. J. K. Ratnadhariya, Principal, HGCE, Vahelal,

for his valuable help and encouragement. I am very much obliged to Smt. Sangita Raje,

Trustee & Vice Chairman Alpha College of Engineering & Technology, Khatraj and Dr.

Santosh S. Kolte, Principal, Alpha College of Engineering & Technology, Khatraj, for their

kind cooperation and support to me at every stage of my research work.

I wish to acknowledge an everlasting debt of gratitude to my beloved family members

and my husband for the countless support and continuous motivation during my work.

Without their blessings and encouragement, it would not have been possible for me to

complete my research work.

Finally, my deepest thanks to all those who have helped me directly or indirectly

in the fruitful completion of my research work.

Yoginibahen D. Vashi

xiii

Contents

ABSTRACT ......................................................................................................................... xi

Acknowledgment ................................................................................................................ xii

List of Symbols .................................................................................................................. xvi

List of Figures .................................................................................................................. xviii

List of Tables .................................................................................................................... xxii

1. General Introduction ...................................................................................................... 1

1.1 Introduction ............................................................................................................. 1

1.2 Summary of the thesis ............................................................................................. 2

1.3 A brief description on the state of the art of the research topic .............................. 3

1.4 Definition of the Problem ........................................................................................ 5

1.5 Objective of the work .............................................................................................. 6

1.6 Original contribution by the thesis .......................................................................... 6

1.7 Methodology of Research and Results/Comparisons ............................................. 6

1.8 Achievements with respect to objectives ................................................................ 8

2. Basic Concepts ................................................................................................................. 9

2.1 Introduction ........................................................................................................... 9

2.1.1 Fluid .................................................................................................................. 9

2.1.2 Density .............................................................................................................. 9

2.1.3 Viscosity ........................................................................................................... 9

2.1.4 Newtonian fluid .............................................................................................. 11

2.1.5 Non-Newtonian fluid ...................................................................................... 11

2.1.6 Couple stress fluid .......................................................................................... 11

2.1.7 Porosity ........................................................................................................... 11

2.1.8 Permeability .................................................................................................... 11

2.1.9 Darcy’s law ..................................................................................................... 11

2.2 Magnetic Parameters .......................................................................................... 12

2.2.1 Magnetic field ................................................................................................. 12

2.2.2 Magnetic field strength ................................................................................... 12

2.2.3 Magnetization: (Intensity of Magnetization) .................................................. 12

2.2.4 Magnetic Susceptibility .................................................................................. 12

2.2.5 Permeability of Free Space ............................................................................. 13

2.3 Concept of Ferrofluid ......................................................................................... 13

2.3.1 Fundamental equations of Neuringer- Rosensweig model for ....................... 13

Ferrofluids Lubrication ................................................................................................. 13

xiv

2.4 Surface roughness ............................................................................................... 14

2.4.1 Transverse roughness design .......................................................................... 15

2.4.2 Longitudinal roughness design ....................................................................... 15

2.5 Basic Equations from Fluid Dynamics .............................................................. 16

2.5.1 Equation of Continuity ................................................................................... 16

2.5.2 Equation of Continuity in vector form ........................................................... 16

2.5.3 Equation of Continuity in the cylindrical form .............................................. 16

2.5.4 Navier-Stokes Equation .................................................................................. 17

2.5.5 Generalized Reynolds equation ...................................................................... 17

2.5.6 Derivation of Generalized Reynolds type equation for couple ...................... 18

stress fluid based parallel stepped plates ........................................................ 18

2.5.7 Generalized Reynolds equation for couple stress fluid-based ........................ 24

circular stepped plates .................................................................................... 24

2.5.8 Generalized Reynolds equation for doubled layered porous .......................... 26

plates ............................................................................................................... 26

3. Ferrofluid Lubrication of Rough Porous Parallel Stepped Plates with Couple

Stress .............................................................................................................................. 28

3.1 Introduction .......................................................................................................... 28

3.2 Analysis ................................................................................................................ 30

3.3 Results and discussion ......................................................................................... 33

3.4 Conclusion .......................................................................................................... 52

4. Performance of Ferrofluid Based Longitudinally Rough Porous Parallel Stepped

Plates with Couple Stress ............................................................................................. 54

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

4.2 Analysis ................................................................................................................ 55

4.3 Result and Discussion .......................................................................................... 59

4.4 Conclusion ........................................................................................................... 71

5. Influence of Ferrofluid Based Doubled Layered Porous Conical Bearing with two

Different Forms of Transverse Roughness ................................................................. 72

5.1 Introduction .......................................................................................................... 72

5.2 Analysis ................................................................................................................ 74

5.3 Results and discussion ......................................................................................... 79

5.4 Conclusion ........................................................................................................... 88

6. Ferrofluid Lubrication of Double Layered Rough Circular Plates with Slip

Velocity ........................................................................................................................... 90

6.1 Introduction .......................................................................................................... 90

xv

6.2 Analysis ................................................................................................................ 91

6.3 Results and Discussions ....................................................................................... 94

6.4 Conclusion ........................................................................................................... 99

7. Ferrofluid Based Longitudinally Rough Porous Circular Stepped Plates in the

Existence of Couple Stress .......................................................................................... 100

7.1 Introduction ........................................................................................................ 100

7.2 Analysis .............................................................................................................. 102

7.3 Result and Discussion ........................................................................................ 106

7.4 Conclusion ......................................................................................................... 116

8. General Conclusions and Future Scope of The Work ............................................. 117

8.1 General Conclusions .......................................................................................... 117

8.2 Future Scope of the Work .................................................................................. 118

References ......................................................................................................................... 119

List of Publications .......................................................................................................... 126

xvi

List of Symbols

a radius of the circular plate (mm)

b width of the bearing

h mean fluid film thickness (mm)

0h intial film thickness (mm)

1h maximum film thickness (mm)

2h minimum film thickness (mm)

sh devition from mean film thickness

0, ,h h V• •

squeeze velocity of bearing surface

, iH H total film thickness

0H thickness of the porous facing

H external magnetic field vector

H nondimensional mean film thickness

2

1

h

h

1H the thickness of the inner layer of the porous plate (mm)

2H the thickness of the outer layer of the porous plate (mm)

KL or KR position of the step ( )10 K

l couple stress parameter

l non dimensional couple stress parameter

2

2

h

l

L length of the bearing

M magnetization vector

p pressure distribution in the fluid film region (N/m2)

1p pressure in the fluid film region ( )0 x KL or ( )0 r KR

2p pressure in the fluid film region ( )KL x L or ( )KR r R

( ), ,q u v w= fluid velocity in the film region.

R radius of the circular plate

r radial coordinate

s slip parameter

xvii

*s nondimensional slip velocity

W load carrying capacity (N)

W nondimensional load capacity

variance (mm)

* nondimensional variance

skewness (mm)

* nondimensional skewness

couple stress constant of the lubricant

dynamic viscosity of lubricant (N.S/m2)

magnetic susceptibility

0 permeability of free space (N/A2)

density of fluid

standard deviation (mm)

* nondimensional standard deviation

permeability of the porous facing (m2)

1 the permeability of inner layer (m2)

2 the permeability of outer layer (m2)

porosity

1 porosity of inner layer

2 porosity of outer layer

xviii

List of Figures

FIGURE 2.1: Velocity distribution near a solid boundary ............................................................................ 10

FIGURE 3.1: Configuration of rough parallel stepped plates ............................................ 31

FIGURE 3.2 Profile of W for the combination of * and K ............................................. 34

FIGURE 3.3 Profile of W for the combination of * and *H ........................................... 34

FIGURE 3.4 Profile of W for the combination of * and * ........................................... 35

FIGURE 3.5 Profile of W for the combination of * and * ......................................... 35

FIGURE 3.6 Profile of W for the combination of * and * .......................................... 35

FIGURE 3.7 Profile of W for the combination * and ................................................ 36

FIGURE 3.8 Profile of W for the combination of * and l ........................................... 36

FIGURE 3.9 Profile of W for the combination of K and *H .......................................... 38

FIGURE 3.10 Profile of W for the combination of K and ........................................ 39

FIGURE 3.11 Profile of W for the combination of K and * .......................................... 39

FIGURE 3.12 Profile of W for the combination of K and * ......................................... 39

FIGURE 3.13 Profile of W for the combination of K and .......................................... 40

FIGURE 3.14 Profile of W for the combination of K and l ........................................... 40

FIGURE 3.15 Profile of W for the combination of *H and * ........................................ 42

FIGURE 3.16 Profile of W for the combination of *H and * ........................................ 42

FIGURE 3.17 Profile of W for the combination of *H and * ......................................... 43

FIGURE 3.18 Profile of W for the combination of *H and ......................................... 43

FIGURE 3.19 Profile of W for the combination of *H and l ........................................ 43

FIGURE 3.20 Profile of W for the combination of * and * ........................................ 45

FIGURE 3.21 Profile of W for the combination of * and * ........................................ 45

FIGURE 3.22 Profile of W for the combination of * and .......................................... 46

FIGURE 3.23 Profile of W for the combination of and l .......................................... 46

FIGURE 3.24 Profile of W for the combination of * and * ......................................... 48

FIGURE 3.25 Profile of W for the combination of * and ........................................ 48

FIGURE 3.26 Profile of W for the combination of * and l ........................................... 48


FIGURE 3.28 Profile of W for the combination of * and l .......................................... 50


FIGURE 4.1 Configuration of longitudinally rough porous parallel stepped plates .......... 56

FIGURE 4.2 Profile of W for the combination of * and K ............................................. 60

FIGURE 4.3 Profile of W for the combination of * and *H .......................................... 60

xix

FIGURE 4.4 Profile of W for the combination of *μ and * ............................................ 60

FIGURE 4.5 Profile of W for the combination of * and * ............................................ 61


FIGURE 4.7 Profile of W for the combination of * and ........................................... 61

FIGURE 4.8 Profile of W for the combination of * and l ............................................. 62

FIGURE 4.9 Profile of W for the combination of K and H ......................................... 62

FIGURE 4.10 Profile of W for the combination of K and ....................................... 62

FIGURE 4.11 Profile of W for the combination of K and ....................................... 63


FIGURE 4.13 Profile of W for the combination of K and ........................................... 63

FIGURE 4.14 Profile of W for the combination of K and l ........................................ 64

FIGURE 4.15 Profile of W for the combination of H and ...................................... 64

FIGURE 4.16 Profile of W for the combination of H and ....................................... 65


FIGURE 4.18 Profile of W for the combination of H and ........................................ 65

FIGURE 4.19 Profile of W for the combination of H and l ......................................... 66

FIGURE 4.20 Profile of W for the combination of and ........................................ 66

FIGURE 4.21 Profile of W for the combination of and ....................................... 66

FIGURE 4.22 Profile of W for the combination of and ......................................... 67

FIGURE 4.23 Profile of W for the combination of and l ........................................... 67

FIGURE 4.24 Profile of W for the combination of and .......................................... 67

FIGURE 4.25 Profile of W for the combination of and ........................................... 68

FIGURE 4.26 Profile of W for the combination of and l ........................................... 68


FIGURE 4.28 Profile of W for the combination of and ............................................ 69

FIGURE 5.1. Configuration of rough conical bearing ....................................................... 75



FIGURE 5.4 Profile of W for the combination of * and ............................................. 80

FIGURE 5.5 Profile of W for the combination of * and 1 ........................................... 80

FIGURE 5.6 Profile of W for the combination of * and ........................................... 81

FIGURE 5.7 Profile of W for the combination of * and ............................................. 81



FIGURE 5.10 Profile of W for the combination of and 1 .......................................... 82

FIGURE 5.11 Profile of W for the combination of and 2 ......................................... 83


*

2

xx

FIGURE 5.13 Profile of W for the combination of * and 1 .......................................... 84

FIGURE 5.14 Profile of W for the combination of * and 2 ........................................ 84

FIGURE 5.15. Profile of W for the combination of and ........................................ 84

FIGURE 5.16 Profile of W for the combination of and 1 ........................................... 85

FIGURE 5.17 Profile of W for the combination of and 2 .......................................... 85


FIGURE 5.19 Profile of W for the combination of 1 and ......................................... 86

FIGURE 5.20 Profile of W for the combination of 1 and 2 ......................................... 86


FIGURE 6.1 Configuration of the doubled layered circular plates .................................... 91



FIGURE 6.4. Profile of W for the combination of * and 1 ......................................... 95

FIGURE 6.5 Profile of W for the combination of * and 2 .......................................... 95

FIGURE 6.6 Profile of W for the combination of *

1

s and ........................................ 96

FIGURE 6.7 Profile of W for the combination of 1

s and ......................................... 96


1

s and ......................................... 96


1

s and 2 ........................................ 97


FIGURE 6.11 Profile of W for the combination of and 2 ......................................... 98

FIGURE 6.12 Profile of W for the combination of * and ......................................... 98

FIGURE 6.13 Profile of W for the combination of * and 2 ......................................... 98

FIGURE 6.14 Profile of W for the combination of * and 2 ........................................ 99

FIGURE 7.1 The physical geometry of longitudinally rough circular stepped plates ..... 102

FIGURE 7.2 Profile of W for the combination of and K .......................................... 106

FIGURE 7.3 Profile of W for the combination of and * ........................................ 106

FIGURE 7.4 Profile of W for the combination of and * ......................................... 107

FIGURE 7.5 Profile of W for the combination of and .......................................... 107






xxi

FIGURE 7.11 Profile of W for the combination of K and ......................................... 109

FIGURE 7.12 Profile of W for the combination of K and l ......................................... 110

FIGURE 7.13 Profile of W for combination of H and ............................................ 110



FIGURE 7.16 Profile of W for the combination of H and ........................................ 111

FIGURE 7.17 Profile of W for the combination of H and l ........................................ 112



FIGURE 7.20 Profile of W for the combination of and ......................................... 113

FIGURE 7.21 Profile of W for the combination of and l ......................................... 113

FIGURE 7.22 Profile of W for the combination of and ...................................... 114

FIGURE 7.23 Profile of W for the combination of and l ......................................... 114

FIGURE 7. 24 Profile of W for the combination of and ........................................ 115

FIGURE 7. 25 Profile of W for the combination of and l ........................................ 115

xxii

List of Tables

TABLE 3.1 Distribution of W for the combination of * and K ....................................... 36

TABLE 3.2 Distribution of W for the combination of * and *H .................................... 37

TABLE 3.3 Distribution of W for the combination of * and * ..................................... 37

TABLE 3.4 Distibution of W for the combination of * and * ...................................... 37

TABLE 3.5 Distribution of W for the combination of * and * .................................... 37

TABLE 3.6 Distibution of W for the combination of and ........................................ 38

TABLE 3.7 Distribution of W for the combination of * and l .................................... 38

TABLE 3. 8 Distribution of W for the combination of K and *H .................................. 40

TABLE 3.9 Distribution of W for the combination of K and ...................................... 41

TABLE 3.10 Distribution of W for the combination of K and * .................................... 41

TABLE 3.11 Distribution of W for the combination of K and * .................................... 41

TABLE 3.12 Distribution of W for the combination of K and ................................... 41

TABLE 3.13 Distribution of W for the combination of K and l .................................... 42

TABLE 3.14 Distibution of W for the combination of *H and * ..................................... 44

TABLE 3.15 Distibution of W for the combination of *H and * ................................... 44

TABLE 3.16 Distribution of W for the combination of *H and * ................................... 44

TABLE 3.17 Distribution of W for the combination of *H and .................................. 44

TABLE 3. 18 Distibution of W for the combination of *H and l ................................... 45

TABLE 3.19 Distribution of W for the combination of * and * .................................... 46

TABLE 3.20 Distribution of W for the combination of * and * ................................. 47

TABLE 3.21 Distribution of W for the combination of * and ..................................... 47

TABLE 3.22 Distribution of W for the combination of * and l ..................................... 47

TABLE 3.23 Distribution of W for the combination * and * ..................................... 49

TABLE 3.24 Distribution of W for the combination of * and .................................. 49

TABLE 3.25 Distribution of W for the combination of * and l .................................... 49

TABLE 3. 26 Distribution of W for the combination of * and ................................... 51

TABLE 3.27 Distribution of W for the combination of * and l ...................................... 51

TABLE 3.28 Distribution of W for the combination of and l .................................... 51

TABLE 3.29 Distribution of W and W

R for various values of H and l ....................... 52


R for various values of K and l ........................ 52


R for distinct values of H and l ........................ 69


R for distinct values of K and l ........................ 70

xxiii

TABLE 5.1 Change in W with regards to different values of ...................................... 87

TABLE 5.2. Change in W with regards to different values of ...................................... 88

TABLE 5.3 Change in W with regards to different values of ...................................... 88

TABLE 7.1 Change in W and W

R for distinct values of K and l ............................... 115


R for distinct values of H and l .............................. 116

1

CHAPTER 1

1. General Introduction

1.1 Introduction

The present thesis is devoted to study the impact of various types of bearing on load capacity.

In this theoretical study mathematical model has been established for various types of

squeeze film bearing systems such as circular, parallel stepped, conical, circular parallel

stepped. Ferrofluid is used as a lubricant in these bearings. Tribology is one of the most

important subject dealing with friction, wear, and lubrication. If we could control and reduce

main constituents friction and wear of tribology, automatically it increased the service life

of the apparatus. This, is reduce cost. The identification of tribological problems and their

solutions can increase to significant savings. To reduce friction, lubrication works as a key

role. The selection of suitable lubricant in the machine can extend the machine’s life period

as well.

In recent years, extensive investigation has been made on the study of the influence of

ferrofluid lubrication on bearing performance. This field has gained a wide range of devotion

due to its extensive use in technical purposes like dynamic sealing, heat dissipation, damping

and medical uses like drug targeting, hyperthermia, and cell separation. In recent years,

influence of surface roughness and its impact on machine design have been vital features

that have been widely studied. Some approaches have been recommended to study the

consequence of surface roughness on the bearing performance. Due to the random structure

of the surface roughness, a stochastic model for the study of hydrodynamic lubrication has

been developed by (Christensen & Tonder, 1969a, 1969b, 1970a).

So, the present thesis purposes to examine the joint effect of ferrofluid and surface

roughness on various types of porous bearings. With appropriate boundary conditions, the

related modified Reynolds equation is solved to develop pressure in the film region

General Introduction

2

which gives the load-bearing capacity. Obtained results are presented in graphical forms as

well as tabular forms. In the presence of ferrofluid, the system shows better performance.

1.2 Summary of the thesis

The work has been summaries in the form of various chapters. The thesis consists of seven

chapters. The first chapter introductory in nature and contains motivation for the study.

Chapter II represents the brief discussion on the basic need of lubrication theory and

tribology. It includes properties of the fluid, classification of fluids and derivation of basic

governing equations.

A theoretical study of ferrofluid lubrication of rough porous parallel stepped plates

with the presence of couple stress effect is analyzed in chapter III. The expression for film

pressure and load-bearing capacity has been found as a function of various parameters and

deliberated from different viewpoints. It is noted from the study that the bearing’s load

capacity is enhanced due to the magnetic effect. Roughness and porosity affect bearing’s

load capacity adversely, but this contrary influence can be compensated up to a certain level

with a suitable range of couple stress parameter and magnetic parameter.

Chapter IV deals with the performance of ferrofluid based longitudinally rough

porous parallel stepped plates with couple stress. Obtained results are compared with

conventional lubricant based bearing system. The graphical and tabular representation

emphasizes that the combined impact of magnetization and couple stress is to boost the load

bearing capacity irrespective of the circumstances.

Chapter V presents the influence of a doubled layered porous conical bearing with

two different forms of transverse roughness. Also, a comparison is made between two

roughness patterns. Computed results are presented graphically with regards to various

parameters. It is found that load bearing capacity enhances due to doubled layered plates.

Chapter VI makes an effort to study the combined influence of slip velocity and

surface roughness for double layered porous circular plates with magnetic fluid. The

influence of slip velocity is governing by Beavers and Joseph’s slip model. The results

presented in the graphical forms establish that the magnetic parameters provide a limited

extent in holding the contrary influence of roughness, porosity, and slip velocity. Though,

the condition improves when negatively skewed roughness occurs. But any kind of

development in the bearing performance, the slip has to be kept at a minimum level even

though variance (-ve) is involved.

1.3 A brief description on the state of the art of the research topic

3

Chapter VII analyzes the performance of rough porous circular stepped plates

lubricated with ferrofluid. Neuringer–Roseinweig model has been employed for magnetic

fluid. Bearing surface roughness has been calculated using the stochastic theory given by

Christensen and Tonder. Stokes microcontinum theory has been employed for couple stress

influence. According to the graphical and tabular results obtained, the influence of ferrofluid

lubrication joint with the couple stress impact improves the load capacity of bearing

compared to couple stress fluid-based bearing system.

Chapter VIII covers the over all conclusion and future scope of the work.

On the whole, the present study investigates the performance of ferrofluid based

squeeze film lubrication in different types of rough porous bearing geometries with couple

stress effect.

1.3 A brief description on the state of the art of the research topic

Through the past hundred years of the investigative feature of tribology, a large number of

progress with regards to the investigation, study, and improvements of bearings have been

carried out.The field of tribology has grown an independent position. These advancements

being recognized in the numeral of books. Some of these books become very famous and

are used as reference text is (Pinkus & Sternlitcht, 1961; Tipei, 1962; Cameron,1981;

Majumdar, 1986; Hamrock, 1994; Hirani, 2016).

A scientific approach to friction is given by Leonardo Da Vinci [1452-1519]. He has

derived the basic laws of friction and presented the idea of the coefficient of friction as the

fraction of the friction force to a normal load. A theoretical study of lubrication of bearings

was made by (Obsborne Reynolds ,1886) and he derived a very well-known general equation

for fluid film lubrication known as Reynolds equation.

Porous bearing is used very widely in many devices such as vacuum cleaners,

extractor fans, motorcar starters, hairdryer, etc. They are also used in business machines,

farm and construction equipment, and aircraft automotive accessories. In addition, the

porous bearing can work hydrodynamically longer short of maintenance and steadier than

conventional bearing. Also, in these bearings’ friction is less as associated with conventional

bearings. Over the ancient spans, an extensive number of theoretic models have been

proposed on the performance features of the porous bearings by numerous investigators. The

hydrodynamic model of porous journal bearing based on the Darcy model was investigated


4

initially by (Morgan & Cameron, 1957). Prakash and VIJ (1973a) analyzed the performance

of various porous plates like circular, conical, truncated conical, elliptical, recantangular,

etc. and compared with conventional lubricant based bearing system. Uma Srinivasan

(1977a) has almost extended above work by considering doubled layered porous plates.

Cusano (1972) conducted the study of double layered porous bearing with infinite width.

Prakash and VIJ (1976) studied the influence of a rotating porous annular disk with velocity

slip effect. Wu H (1978) has made analysis of porous squeeze films. Verma (1983)

investigated the influence of a doubled layered porous slider bearing. Xin and Ming (1985)

have made a study for porous bearing including the theoretical and experimental aspects.

Representation of surface roughness is a significant feature in applications including

friction, wear, and lubrication. The scrutiny of the influence of surface roughness on

hydrodynamic lubrication of different bearing systems has been a focus of developing

attention, because, in reality, most of the bearing surfaces are not smooth. The bearing

surface tends to be rough after having some run-in and wear. Christensen and Tonder (1969a,

1969b,1970a,1970b,1971) made a comprehensive model using polynomial probability

distribution function for bearing’s surface roughness. Christensen et al. (1975) derived the

generalized Reynolds equation with the stochastic approach and also gave its applications.

Prakash and Tiwari (1983) studied the influence of roughness in porous circular squeeze

plates considering arbitrary wall thickness. In order to increase the ability of the bearing

performances many theoretical and experimental types of research have been carried out on

the design point of view of bearing as well as lubricating substances. One of the main

inventions of the lubricant is the development of ferrofluid and associated progress.

Neuringer–Rosensweig (1964) model describes the basics hydrodynamic equations leading

the flow of magnetic fluid. They focused on the influence of the magnetic body force on a

paramagnetic fluid characterized by asymmetric Newtonian stress tensor and considered

thermo-mechanical phenomena in this model. With the advent of ferrofluids by

(Rosensweig,1965), several applications in different areas like in sensors, sealing devices,

cleaning apparatus, damper, spindle motor, etc. (Mehta & Upadhyay,1999; Uhlmann et al.,

2002; Scherer & Figueiredo Neto,2005) are found owing to discriminate qualities of ferrofluid.

Dinesh Kumar et. al. (1992) studied the influence of ferrofluid on spherical and conical

bearings using perturbation analysis. Prajapati (1995) studied the effect of magnetic fluid on

porous squeeze film bearings. Bhat and Deheri (1993) made a theoretical study on squeeze

film based curved porous circular disk lubricated with ferrofluid. Shah (2003) analyzed the

ferrofluid lubrication in step bearing. Numerous researchers have analyzed the effect of

1.4 Definition of the Problem

5

surface roughness with the existence of ferrofluid lubrication, for various bearing

geometries. (Patel et al., 2011; Gupta & Deheri,1996; Patel et al., 2008; Patel & Deheri,

2007; Patel & Deheri, 2016a ; Andhariya & Deheri, 2011; Patel & Deheri, 2016b ; Shimpi

& Deheri, 2014). All these studies revealed that there is a substantial increase in load due to

magnetic fluid compared to conventional lubricant case.

Now a day it is well known that the use of Newtonian fluids mixed with additives

introduces a development in the bearing performances as related to the Newtonian lubricants.

In many of these lubricants, the additives of excessive molecular weight polymers exist as a

kind of viscosity index improvers. Key benefits of base oils of high viscosity index are an

extremely consistent component of machine parts in a wide range of working temperatures,

a lengthier life and good reply to additives (Ariman et al.,1974). In fact, owing to the

existence of additives a nonlinear relation is created between the shear stress and strain rate.

Stokes (1966) proposed a microcontinuum theory for the couple stress fluids. This theory is

the generalization of traditional Newtonian fluid law and it deals with the polar effects like

the couple stresses, body couples, and asymmetric tensors. Li Chu (2004) established the

generalized Reynolds equation for thin-film lubrication with rheological impact of couple

stress fluids. Numerous researchers have analysed the impact of couple stress using Stokes

micro-continuum theory for various types of bearing systems (Elkouh & Yang, 1991; Lin,

1998; Bujurke et al., 1990; Lin et al., 2006; Guha, 2004; Ramnaiah, 1966; Ramanaiah &

Sarkar, 1978; Ramanaiah & Dubey, 1975; Maiti, 1973; Naduvinamani & Siddangouda,

2007; Naduvinamani & Siddangouda, 2009; Biradar, 2012; Biradar, 2013). All the above

studies discovered the importance of non-Newtonian fluid in squeeze films and presented

that this non-Newtonian fluid contributed to improved performance in hydrodynamic

lubrication compared to Newtonian lubricant.

1.4 Definition of the Problem

The present effort is made for the theoretic study of surface roughness and ferrofluid

lubrication for various porous bearing geometries. With the traditional principles of

hydrodynamic lubrication, the Generalized Reynolds type equation is solved for parallel

stepped plates, circular plates, and conical plates bearing. The goal of the work is to observe

the influence of ferrofluid with couple stress on bearing’s load capacity. The impact of

transverse and longitudinal surface roughness is studied with roughness parameters like

standard deviation, variance, and skewness.


6

1.5 Objective of the work

The aim of research in the field of tribology is to lessen and remove the losses developing

from friction and wear at all stages of technology which includes the rubbing of surfaces.

Investigation in Tribology directs to larger efficiency, improved performance, fewer

collapses and substantial savings.

The key purpose of this investigation is to analyze the combined influence of magnetic fluid

and surface roughness on bearing’s load capacity by using (Neuringer Rosenweig, 1964)

model and (Christensen & Tonder, 1969a, 1969b, 1970a). Various types of bearing

geometries are considered for the study like parallel stepped plates, circular plates, conical

plates, and circular stepped plates, etc. Also, the aim is to find a closed-form solution for

film pressure and bearing’s load capacity.

1.6 Original contribution by the thesis

The original contribution by the thesis is based on mathematical modeling of parallel stepped

plates, conical plates, circular plates and circular stepped bearing which analyses:

▪ The combined influence of roughness and ferrofluid on porous parallel stepped plates

with couple stress effect.

▪ Influence of double layered porous conical plates with two different patterns of

transverse roughness.

▪ Impact of slip velocity and surface roughness on ferrofluid based double layered porous

circular plates.

▪ Performance of longitudinally rough porous circular stepped plates with the existence

of ferrofluid and couple stress effect.

The analytic solution of such problems is obtained and results are plotted graphically.

Comparisons are given between the ferrofluid based bearing system and the conventional

lubricant based bearing system.

1.7 Methodology of Research and Results/Comparisons

The inspiration for the current work arises from the observation of the occurrence of

lubrication phenomenon in numerous applications like automotive and aircraft engines,

bearings, dampers, gears, clutches, turbomachinery, and skeletal joints. Due to these

1.7 Methodology of Research and Results/Comparisons

7

extensive applications of lubrication, many kinds of research work has been carried out on

the phenomenon by numerous investigators from distinct viewpoints.

Following traditional assumptions of hydrodynamic lubrication are made for study.

▪ The incompressible lubricant with constant velocity and constant viscosity is

considered.

▪ The flow of the lubricant is laminar and steady. The fluid properties should not be

changed with regard to time.

▪ The fluid film thickness is deliberated very small in comparison to the dimensions of

the bearings

▪ Body forces are ignored, i.e. there are no outer fields of force acting on the fluid.

▪ The porous region is supposed to be homogeneous and isotropic.

Under the above assumptions of hydrodynamic lubrication, according to the Stokes

microcontinum theory for couple stress fluid the generalized Reynolds type equation for

parallel stepped plates is derived with no-slip boundary conditions for the smooth bearing is

given by (Biradar,2012)

( )12

,i i

dp Vx

dx G H l

−=

(1.1)

where,

( ) 3 2 3, 12 242

ii i i i

HG H l H l H l tanh

l

= + +

Equation (1.1) is modified to obtain the surface roughness effect and magnetization effect.

The bearing surface’s roughness effect is obtained on the basis of (Christensen & Tonder,

1969a, 1969b, 1970a) model for hydrodynamic lubrication of rough surfaces. Transverse

and longitudinal roughness with nonzero mean has been considered for the study. Fluid in

the film region is described by (Neuringer–Rosensweig, 1964) model for ferrofluid

lubrication. A porous surface is deliberated because of getting the benefit of the self-

lubricating property. Porosity is governed by Darcy’s law.

The modified Reynolds equation for double layered porous plates is given by

(Srinivasan, 1977a)

( ) ( )3 31 1 2 1 1 2 212 12 12 12 6 122 h

p p dhh H H h H H U V

x x z z dx

+ + + + + = +

(1.2)


8

The modified Reynolds equation for double layered porous plates in polar coordinates is

given by

( ) ( )3 31 2 1 22

1 112 12 12 12

6 12

1 2 1 2θ θ

U Vθ

p ph H H r h H H r

r r r r

dh sinθ dhcosθ

dr r d

+ + + + + =

− +

(1.3)

The fluid flow becomes axisymmetric in the case of circular plates so equation (1.3) turns

out to be

31 1 2 2

121

12 12

dh

d dp dtrr dr dr h H H

=

+ +

(1.4)

For conical plates bearing modified Reynolds equation is

3 31 1 2 2

121

12 12

dhsin

d dp dtxx dx dx h sin H H

=

+ +

(1.5)

In the present study equations (1.4) and (1.5) are modified to obtain the surface roughness

influence in the presence of ferrofluid lubrication. Closed-form solutions are obtained for

film pressure and bearing’s load capacity in terms of various parameters and presented

graphically.

1.8 Achievements with respect to objectives

Deploying a theoretic approach study of solving squeeze film flow problems is carried out.

The generalized Reynolds equation is solved with no-slip boundary conditions and modified

to achieve our objectives. In all the present studies modified Reynolds equation leading the

pressure distribution is averaged with regards to the roughness parameter. The equation for

dimensionless load bearing capacity is found in the form of roughness parameters,

magnetization parameter, and couple stress parameter. Graphical and tabular results indicate

improved performance due to ferrofluid lubrication compared to conventional lubricant.

.

9

CHAPTER 2

2. Basic Concepts

2.1 Introduction

In this chapter several definitions, which are essential for the consequent study, are

deliberated. It also contains basic equations of fluid dynamics and derivation of modified

Reynolds equation for different bearing geometry, so it will offer a base for the subsequent

chapters of the thesis.

2.1.1 Fluid

A substance that is capable of flowing is called a fluid.

2.1.2 Density

Density is defined as the ratio of mass to volume of a fluid. It is represented by a symbol

. The SI unit of density is kg/meter3

Mathematically, density is stated as Mass of fluid

volumeof fluid=

2.1.3 Viscosity

The property of the fluid which resists the movement of one layer of the fluid over an

alternative adjacent layer of the fluid is known as viscosity.

Figure 2.1 displays the moment of fluid layers.The distance between these two fluid layers

is dy . The velocities of these layers are u and u du+ respectively. The shear stress is cause

Basic Concepts

10

among these layers of fluid due to the viscosity and relative velocity. This shear stress is

proportional to the rate of change of u (velocity) with regard to y . Symbolically it is

represented by .

FIGURE 2.1: Velocity distribution near a solid boundary

The above statement can be express mathematically as,

du

dy

du

dy =

Where the proportionality constant is known as the coefficient of dynamic viscosity or

viscosity.

From above relation, we have

du

dy

=

2.1 Introduction

11

2.1.4 Newtonian fluid

A real fluid that obeys the Newton’s law of viscosity is identified as the Newtonian fluid.

2.1.5 Non-Newtonian fluid

A real fluid that does not obey the Newton’s law of viscosity is identified as non-Newtonian

fluid.

2.1.6 Couple stress fluid

When we blend additives in the fluid, the forces which are existing in the fluid resists the

forces of additives. This obstruction builds a couple force and so couple stress is made in the

fluid. This kind of fluid is recognized as a couple stress fluid. The growing use of fluids

having microstructure such as additives, granular matter or long-chained polymer

suspensions has been accentuated owing to the growth of the current machine apparatus. The

traditional Newtonian theory will not precisely define the rheological behavior of lubricants

mixed with several additives. A numeral of microcontinuum theories has been projected by

(Stokes,1966).

2.1.7 Porosity

Porosity determines the measure of void spaces in a porous material. It is the fraction of the

volume of void spaces to the total volume of the material.

2.1.8 Permeability

It determines the measure of the flow conductivity in the porous material. The SI unit of

permeability is m2. A practical unit of permeability is Darcy.

2.1.9 Darcy’s law

In the year 1856 Darcy has introduced the first governing equation for the flow of fluid in a

porous medium. Accordingly, the law is described by the equation

Basic Concepts

12

p= −

V

Where V is known as Darcy velocity, represents porous facing’s permeability. is

described as the coefficient of viscosity and the pressure in the porous region is represented

by p

2.2 Magnetic Parameters

The magnetic property of materials is subject to the degree of magnetization. The magnetic

materials are described by parameters like magnetization, magnetic susceptibility, and

magnetic permeability. The essential magnetic parameters which are utilized to describe the

magnetic materials are as follows.

2.2.1 Magnetic field

A magnetic field is an area around a magnet where its magnetic effect is experienced.

2.2.2 Magnetic field strength

Magnetic field strength H is the force employed by the magnetic field at a given point in

the field. It is measured in amperes per meter (A/m).

2.2.3 Magnetization: (Intensity of Magnetization)

It is the extent to which a specimen is magnetized when placed in a magnetizing field. The

unit of magnetization is Am-1

2.2.4 Magnetic Susceptibility

It is denoted by the symbol . It is the ratio of the magnetization M to the applied

magnetic field strength H .

Mathematically, it is expressed as

2.3 Concept of Ferrofluid

13

M=

H

2.2.5 Permeability of Free Space

The free space permeability is a physical constant. It is denoted by the symbol 0 . Its

value is

70 4 10−= N/A

2.3 Concept of Ferrofluid

The ferrofluid is a suspension of solid magnetic particles of subdomain size in a liquid

carrier. Ferrofluids are prepared by using different types of base fluids, surfactants, and

particles. For example, a ferric oxide particle coated with surfactant antimony and suspended

in a base fluid diester. The mean diameter of the particle is varying between 3 to 15nm.

Ferrofluid is a liquid that turns into intensely magnetized in the existence of the external

magnetic field.

2.3.1 Fundamental equations of Neuringer- Rosensweig model for

Ferrofluids Lubrication

Neuringer-Rosenswein (1964) intended a model to define the stable flow of ferrofluids in

the being of gradually varying magnetic fields. The model involves the following equations:

( ) ( )20. .ρ q q p q M H = − + + (2.1)

. 0q = (2.2)

0= H (2.3)

M H= (2.4)

( ). 0H M + = (2.5)

Using (2.4)

Basic Concepts

14

( ) ( )0 0. .M H H H =

By making the use of vector identity

( ) ( )1

. . ( )2

H H H H H H = − (2.6)

and supposing the displacement current for electrically non-conducting fluid is insignificant,

therefore 0= H , we get

( ) 2 20.2

ρ q q p q

= − − +

H (2.7)

Equation (2.7) represents that extra pressure 2

0

1

2 H is present into the equation of motion

when ferrofluid is considered as a lubricant.

2.4 Surface roughness

No hard surface is completely smooth on microscopic measure. In other words, all hard

surfaces are rough to some level. In the study of science and technology of tribology, surface

roughness plays a considerable role. Christensen & Tonder (1969a, 1969b, 1970a) have

established the stochastic approach for the hydrodynamic lubrication of rough surface. The

film thickness is observed as a randomly varying quantity so, the surface roughness is

calculated by using the height distribution function. Gaussian distribution is approximated

by a polynomial probability distribution function. In the context of Christensen &

Tonder(1969a, 1969b, 1970a) polynomial probability distribution function is defined as

( )( )

32 2

7

1

35,

32

0 , elsewhere

s ss

c h c h cf h c

− −

=

(2.8)

Additionally, a different form of this kind of polynomial distribution from (Prajapati, 1995)

is

( )( )

22 2

5

2

15,

16

0 , elsewhere

s ss

c h c h cf h c

− −

=

(2.9)

2.4 Surface roughness

15

Where c represents the maximum deviation from the mean level. The deviation from the

mean level sh is stochastic in nature and describes the roughness parameters the non zero

mean ( ) , standard deviation ( ) , skewness ( ) . These roughness parameters are

demarcated by the following relations

( )sE h=

( )22

sE h = −

( )3

sE h = −

Where E is the expectancy operator defined by

( ) ( )s s s sE h h f h dh

−=

Here can assume only positive values while and can assume both negative and

positive values.

On the basis of stochastic theory (Christensen & Tonder, 1969a, 1969b, 1970a) the study is

generally performed for two kinds of roughness designs (viz. transverse and longitudinal) as

follows

2.4.1 Transverse roughness design

In this design, the roughness is supposed to have the form of long, narrow ridges and furrows

running across the direction of sliding.

2.4.2 Longitudinal roughness design

In this design, the roughness is assumed to have the form of long, narrow ridges and valleys

running in the direction of sliding.

Basic Concepts

16

2.5 Basic Equations from Fluid Dynamics

2.5.1 Equation of Continuity

The continuity equation derived on the principle of conservation of mass. The law of

conservation of mass is stated as “Mass can be neither created nor destroyed”.

The most general form of continuity equation in cartesian coordinate is

( ) ( ) ( ) 0u v wt x y z

+ + + =

(2.10)

Equation (2.10) is valid for steady and unsteady flow, uniform and non-uniform flow as

well as compressible and incompressible fluids.

For the steady flow t

becomes zero and hence (2.10) turns into

( ) ( ) ( ) 0u v wx y z

+ + =

(2.11)

Density remains constant for incompressible fluid hence (2.11) turn into

0u v w

x y z

+ + =

2.5.2 Equation of Continuity in vector form

( ) 0qt

+ • =

Where,

q u i v j wk

= + +

2.5.3 Equation of Continuity in the cylindrical form

( )1 1

0v w

rur r r z

+ + =

The above equation is valid for incompressible steady flow.


17

2.5.4 Navier-Stokes Equation

It is established on the basis of conservation of momentum. It can be express in the following

form

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

= − + − + + + + + +

22

3

Dv p v u v w v w u vY

dt y y y x y z z z y x y x

= − + − + + + + + +

22

3

Dw p w u v w w u y w

dt z z z x y z x x z y z y

= − + − + + + + + +

In the above equations, the velocity components in , ,x y z directions are characterized by

, ,u v w whereas p indicates the fluid film pressure. , ,Du Dv Dw

dt dt dt are the components of the

acceleration of the fluid. In the expanded form it can be written as

Du u u u uu v w

dt x y z t

= + + +

The component Dv

dtand

Dw

dtcan also be written in a similar way.

In the Navier-Stokes equation left-hand side term characterize inertia term and right side are

the body forces, pressure gradient, and viscous term.

2.5.5 Generalized Reynolds Equation

Generally, the Reynolds equation leading the fluid film pressure is a second-order nonlinear

partial differential equation, which is formed by using the theory of hydrodynamic

lubrication in equations of motion and continuity as an extension of Navier-Stokes equations.

The development of Reynolds equation for further general cases like rough bearings, porous

bearings or bearing with hydromagnetic lubrication or bearings working with non-

Newtonian or ferrofluid lubrication, etc. is entitled as modified Reynolds equation or

generalized Reynolds equation. This equation is recognized as the fundamental leading

differential equation for the problems of hydrodynamic lubrication.

Basic Concepts

18

( ) ( ) ( )3 3

12 12 2 2

a b a b hu u h w w hh p h p

x x z z x z t

+ + + = + +

(2.12)

In (2.12) left side two terms to define the net flow rates owing to pressure gradients, the first

two terms of the right-hand side of equation define the flow rates due to surface velocities.

These terms are known as Poiseuille and Couette terms respectively.

2.5.6 Derivation of Generalized Reynolds type equation for couple

stress fluid based parallel stepped plates

With the traditional assumption of hydrodynamic lubrication of thin films, the momentum

equation and continuity equation developed by (Stokes, 1966) for the couple stress fluid

yield the form.

Momentum equation

2 4

2 4

u u p

xy y

− =

(2.13)

0p

y

=

(2.14)

Equation of continuity :

0u v

x y

+ =

(2.15)

The associated boundary conditions for the velocity components are given by

On the upper surface y H=

2

20, 0

uu

y

= =

(2.16a)

v V= − (2.16b)

On the lower surface 0y =

2

20, 0

uu

y

= =

(2.17a)

v V = (2.17b)

Where V represents the Darcy velocity component in the y-direction in the porous region.


19

pV

y

= −

Integrating (2.13) two times with respect to y we obtain

2 2

2 2 2

1

2

u u p yAy B

xy l l

− = − + +

(2.18)

Using the boundary condition (2.17a)

At 0y = , 2

20, 0

uu

y

= =

Then (2.18) reduces to

2

10 0 0 0 B

l− = − + +

0B = (2.19)

Using the boundary condition (2.16a) in (2.18)

At y H= , 2

20, 0

uu

y

= =

2

2

10

2

p HAH B

xl

= − + +

2

2

10 0

2

p HAH

xl

= − + +

2

p HA

x

= −

(2.20)

Substitute the value of A and B in (2.18) it reduces to

2 2

2 2 2

1

2

u u p yyH

xy l l

− = − −

(2.21)

Our aim is to find the solution of (2.21). The general solution of (2.21) is obtained by the

rule

( ) complementaryfunction + particular integralu y =

To find a complementary function related characteristics equation is

Basic Concepts

20

2

2

10D u

l

− =

1 2CF

y y

l lc e c e−

= + (2.22)

To obtain the particular integral

( )2

22

2

1 1PI

1

py yH

xlD

l

= − −

−

( )( )2

2 2

1 1PI

1

py yH

xD l

= −

−

( )2 2 4 4 21PI 1 ...

2

pD l D l y yH

x

= + + + −

2 21PI 2

2

py yH l

x

= − +

(2.23)

Hence the general solution of (2.21) is

( ) 2 21 2

1 + 2

2

y y

l lp

u y c e c e y yH lx

− = + − +

(2.24)

To obtain the particular integral we need to find the value of 1c and 2c by making the use of

(2.16a) and (2.17a) in (2.24) we get the following equations

2

1 2 l p

c cx

+ = −

(2.25)

2

1 2 =

H H

l ll p

c e c ex

− + −

(2.26)

By multiplying (2.25) with

H

le and subtracting (2.25) from (2.26) we get

2

2

1

2

H

ll ep

cHx

sinhl

−

= −

(2.27)

Similarly, multiplying (2.25) with

H

le−

and subtracting (2.25) from (2.26) we obtain


21

2

1

1

2

H

ll ep

cHx

sinhl

− −

=

(2.28)

Hence the particular solution of equation (2.24) is obtained as

2 2

2 2

1 11

+ 22

2 2

H H

l l

y y

l l

l e l ep p p

u e e y yH lH Hx x x

sinh sinhl l

−

−

− −

= − − +

22 21

1 1 + 22

2

H y H y

l l l ll p p

u e e e e y yH lH x x

sinhl

− − = − − − − +

22 21

+ 22

2

y H y H y y

l l l ll p p

u e e e e y yH lH x x

sinhl

− −− −

= − − − − +

22 21

+ 22

l p y H y pu sinh sinh y yH l

H x l l xsinh

l

− = − − +

22 22 1

+ 22 2 2

l p y H H pu -2cosh sinh y yH l

H x l l xsinh

l

− = − +

2

2 2

24

1 12 2 + 2

2 22

2 2

y H Hl cosh sinh

p pl lu y yH l

H Hx xcosh sinh

l l

− − = − +

2

2 2

22

1 22

2

2

y Hl cosh

p lu y yH l

Hxcosh

l

− − = + − +

(2.29)

The lubricant’s volume flux is obtained by the relation

0

H

Q b u dy=

Basic Concepts

22

2

2 2

0

22

1 22

2

2

H

y Hl cosh

p lQ b y yH l dy

Hxcosh

l

− − = + − +

33 2

2

0

22

22

2 3 2

2

Hy H

l sinhb p y yl

Q H l yHx

coshl

− = − + − +

3 3 3 2 312 2 3 12 1212 2 2

b p H HQ l tanh H H l H l tanh

x l l

= − + − + −

3 2 312 2412 2

b p HQ H l H l tanh

x l

= − − +

(2.30)

Now we solve (2.15) across the fluid film from 0y = to y H= with (2.16b) and (2.17b)

0

0H u v

dyx y

+ =

By making the use of Leibniz rule of differentiation under integral sign in we get

0 0

0H H

u dy v dyx y

+ =

0 0

H H

u dy v dyx y

= −

By making use of (2.30) we get

0

Hvy

Qb dy

x

= −

0

Hv

Qb

x

= −

3 2 312 24 12 122

H pH l H l tanh V V

x l x

− + = − −

3 2 312 24 12 122

H p pH l H l tanh V

x l x y

− + = − − −

(2.31)

The porous facing thickness 0H is assumed to be small, the (Morgan-Cameron,1957)

approximation gives


23

2

0 2

p pH

yx

= −

(2.32)

Using (2.32) in (2.31) we get the generalized Reynolds type equation turns out to be

( ), 12p

G H l Vx x

= −

(2.33)

Where

( ) 3 2 30, 12 12 24

2

HG H l H l H l tanh

l

= − + +

In the case of axisymmetric equation (2.33) reduces to

( ), 12d dp

G H l Vdx dx

= −

(2.34)

Taking integration of (2.34) with boundary condition

0 at 0dp

xdx

= =

( )12

,

dp Vx

dx G H l

−=

(2.35)

Where

( ) 3 2 30, 12 12 24

2

HG H l H l H l tanh

l

= − + +

When a ferrofluid is used as lubricant then, in the context of (Neuringer-Rosensweing ,1964)

(2.35) is transformed into the form

( )

20 12

2 ,

d Vxp

dx G H l

−− =

H (2.36)

In the present thesis equation (2.36) is extended for transverse roughness structure as well

as longitudinal roughness structure.

Basic Concepts

24

2.5.7 Generalized Reynolds equation for couple stress fluid-based

circular stepped plates

With the traditional assumption of hydrodynamic lubrication of thin films, the momentum

equation and continuity equation developed by (Stokes, 1966) for the couple stress fluid

yield the from

Momentum equation

2 4

2 4

u u p

rz z

− =

(2.37)

0p

z

=

Continuity equation:

( )1

0w

rur r z

+ =

(2.38)

The associated boundary condition for velocity components are

At the upper surface z H=

2

20, 0

uu

y

= =

(2.39a)

w V= − (2.39b)

At the lower surface 0z =

2

20, 0

uu

y

= =

(2.40a)

w w= (2.40b)

Where w is the Darcy velocity component in the z-direction in the porous facing.

pw

z

= −

The solution of (2.37) with boundary conditions (2.39a) and (2.40a) we get


25

2

2 2

22

1 22

2

2

z Hl cosh

p lu z zH l

Hrcosh

l

− − = + − +

(2.41)

Where couple stress parameter is defined by

l =

The lubricant’s volume flux is obtained by the relation

0

2H

Q r u dz =

2

2 2

0

22

1 22 2

2

2

H

z Hl cosh

p lQ r z zH l dz

Hrcosh

l

− − = + − +

3 2 312 242

r p HQ H l H l tanh

r l

= − − +

Now we solve (2.38) across the fluid film from 0y = to y H= , with boundary conditions

w V= − at y H= and w w= at 0y =

We get

3 2 3112 24 12 12

2

H pH l H l tanh r V w

r r l r

− + = − −

3 2 3112 24 12 12

2

H p pH l H l tanh r V

r r l r z

− + = − − −

(2.42)

Using the (Morgan-Cameron,1957) approximation in (2.42) we get

( )1

, 12p

F H l r Vr r r

= −

(2.43)

Where

( ) 3 2 30, 12 12 24

2

HF H l H l H l tanh

l

= − + +

In the case of axisymmetric (2.43) becomes

Basic Concepts

26

( )1

, 12d dp

F H l Vr dr dr

= −

(2.44)

Taking integration of (2.44) with boundary condition

0 at 0dp

rdr

= =

( )6

,

dp Vr

dr F H l

−=

(2.45)

Equation (2.45) is extended in the present thesis for longitudinal roughness design in the

existence of ferrofluid.

2.5.8 Generalized Reynolds equation for doubled layered porous

plates

Generalized Reynolds equation for porous bearing obtained by (Morgan & Cameron,1957)

is

3 31

0

6 12 12h

y

p p dh ph h U V

x x z z dx y=

+ = + +

(2.46)

Prakash and Vij (1973a) extended the above equation by substituting the pressure in the

porous region by average pressure with regard to the bearing-wall thickness. Hence (2.46)

converted to

( ) ( )3 30 012 12 6 12 h

p p dhh H h H U V

x x z z dx

+ + + = +

(2.47)

Uma Srinivasan (1977a) extended (2.48) for doubled layered porous plates. So, modified

Reynolds equation converted to

( ) ( )3 31 1 2 2 1 1 2 212 12 12 12 6 12 h

p p dhh H H h H H U V

x x z z dx

+ + + + + = +

(2.48)

For the plate surfaces with circular boundaries, it is appropriate to make use of the polar

form of (2.49) i.e.


27

( ) ( )3 31 2 1 22

1 112 12 12 12

6 12

1 2 1 2θ θ

p ph H H r h H H r

r r r r

dh sinθ dhU cosθ V

dr r d

+ + + + + =

− +

(2.49)

For squeeze film lubrication (normal approach of nonrotating plates)

Take 0, h

dhU V

dt= = .

For the circular and conical plate, the flow turns into axisymmetric hence (2.48) and (2.49)

converted to

31 1 2 2

121

12 123

dhsinω

d dp dtxx dx dx h sin ω H H

=

+ +

(2.50)

31 1 2 2

121

12 12

dh

d dp dtrr dr dr h H H

=

+ +

(2.51)

In the present thesis (2.50) and (2.51) are extended for transverse roughness design with

ferrofluid as a lubricant.Equation (2.36) is extended in chapter 3 to study the the impact of

transverse roughness.

28

CHAPTER 3

3. Ferrofluid Lubrication of Rough Porous

Parallel Stepped Plates with Couple Stress

3.1 Introduction

This chapter aims to study the behaviour of magnetic fluid-based squeeze film on

transversely rough stepped plates with the influence of couple stress. Using the well-known

stochastic model of Christensen and Tonder the roughness effect has been evaluated. The

magnetic fluid flow model of Neuringer - Roseinweig has been adopted to obtain the

influence of magnetization. The governing Reynolds’ type equation is derived on the basis

of stokes microcontinum theory for couple stress fluid. For the expression of pressure

distribution, the stochastically averaged Reynolds’ type equation is solved. which results in

the calculation of load carrying capacity. The graphical results also presented in tabular form

suggest that although the bearing suffers on account of roughness, the magnetization and

couple stress effect save the situation, as this combination does not allow the load carrying

capacity to fall rapidly.

The squeeze film actions arise from the phenomena of two lubricated surfaces

moving toward each other in the normal direction and produce a positive pressure and hence

sustenance a load. The squeeze film lubrication can be found in bearings, machine tools,

human body joints, rolling elements, IC engines, and gears applications.

The study of non –Newtonian fluid dynamics is of vital importance in connection

with plastic manufacturing, lubricating and movements of biological and geophysical fluids.

The non-Newtonian behavior finds applications in the fields of rotating machinery, computer

storage devices, viscometer, crystal growth processes, and heat and mass transfer, etc.

29

(Hughes, 1963; Bujurke, 1987; Maiti, 1973; Elkouh & Yang 1991). Stokes (1966)

microcontinuum theory has been widely used to compute the influence of couple stresses on

the performance of bearing systems (Ramanaiah & Dubey 1975). Nowadays it is well known

that the use of Newtonian fluids mixed with additives introduces a development in the

bearing characteristics as related to the Newtonian lubricants. In fact, owing to the existence

of additives, a nonlinear relationship is created among the shear stress and strain rate. There

are a number of fluid models considering the non-Newtonian properties of the lubricants

such power law, couple stress, and micropolar fluid. (Guha, 2004; Ramanaiah & Sarkar,

1978; Lin, 1998). Lin et al. (2006) talked about averaged inertia principle for non-Newtonian

squeeze films in wide parallel plates through the couple stress fluid model. The load carrying

capacity was observed to be increased because of the impact of couple stresses as compared

to the Newtonian lubricant case.

Biradar (2013) investigated theoretically the influence of couple stress fluid on

squeezing flow between porous parallel stepped plates. It was discovered that due to the

presence of couple stress effect in the lubricant the load bearing capacity got improved and

reduced the response time as compared to the Newtonian lubricant-based bearing system.

The ferrofluid is colloidal suspensions composed of magnetic particles of subdomain

size in a carrier. The key benefit of ferrofluid lubricants as a substitute for conventional one

is that the former can be reserved at the desired position by an external magnetic field.

Ferrofluids have been found to be used in many technological applications like dynamic

sealing, heat dissipation, damping and medical applications like drug targeting,

hyperthermia, cell separation, etc. The brief study of the above applications can be had from

(Scherer & Figueiredo Neto, 2005). Huang and Wang (2016) prepared a review report on

the progress of ferrofluid lubrication based on the three flow models of Neuringer–

Rosensweig, Shliomis and Jenkins. They have briefly discussed some experimental studies

on ferrofluid lubrication and concluded that over the conventional lubricant ferrofluid have

considerably better friction decline and anti-wear abilities under the external magnetic field.

Patel et al. (2017a) examined experimentally the influence of ferrofluid based hydrodynamic

journal bearing with a different combination of materials.

Due to the random structure of surface roughness, a stochastic approach was

employed to model the surface roughness (Christensen & Tonder, 1969a, 1969b, 1970a).

The stochastic averaging method deployed in the above investigation found its applications

in a number of investigations (Prakash & Tiwari, 1983; Patel et al., 2008; Gupta & Deheri,

1996). Patel et al. (2008, January) analyzed the performance of a squeeze film between

Ferrofluid Lubrication of Rough Porous Parallel Stepped Plates with Couple Stress

30

infinitely long porous rough parallel plates with a porous matrix of variable film thickness

in the presence of a ferrofluid. The ferrofluid lubrication improved the bearing performance

while the composite roughness of the bearing surfaces induced an adversarial influence on

the squeeze film behavior. Siddangouda (2015a) studied the squeeze film characteristics

between parallel stepped plates considering the influence of couple stresses and surface

roughness. It was noticed that the squeeze film characteristics got enhanced for transverse

roughness whereas the bearing suffered owing to the existence of a longitudinal roughness

pattern. Vadher et al. (2008) considered the problem of squeeze films between electrically

conducting rough porous surfaces and electrically conducting lubricant in the presence of a

transverse roughness for a circular shape of the bearing surfaces. The negative influence of

transverse surface roughness was reduced due to magnetization. This positive effect further

enhanced in the case of negatively skewed roughness.

Patel and Deheri (2016a) studied the influence of ferrofluid on a rough parallel plate

slider bearing. They have made a comparison between three ferrofluid flow models.

Regarding the life period of bearing the Shliomis model is good for higher load while the

Neuringer–Rosensweig model may be deployed for the lower load. Shimpi and Deheri

(2012) investigated the effect of deformation and surface roughness for ferrofluid based

rotating porous curve circular plates. The load bearing capacity was found to be decreasing

due to the effect of rotation and deformation, it is improved due to ferrofluid lubrication in

the case of negatively skewed roughness. The squeeze film characteristics for an infinitely

long rough rectangular plate under the presence of a ferrofluid was examined by (Deheri et

al., 2006) this examination built up that the negative effect of surface roughness could be

adjusted to a nominal extent by the ferrofluid lubrication.

So, in this chapter investigation is made to observe the influence of a ferrofluid on

squeeze film between rough stepped plates with couple stress.

3.2 Analysis

Fig. 3.1 indicates the structure of the bearing system wherein, the squeeze film between

parallel stepped plates moving toward each other with normal velocity V is depicted. All

the principle of hydrodynamic lubrication is expected here. The lubricant is an

incompressible ferrofluid based Stoke couple stress fluid.

3.2 Analysis

31

FIGURE 3.1: Configuration of rough parallel stepped plates

With the aid of (Stokes, 1966) microcontinum theory for couple stress fluid,

following the study of (Biradar, 2012) the associated generalized Reynolds’ type equation

for the fluid film pressure is established to be

( )12

,

i

i i

dp Vx

dx G H l= −

(3.1)

where,

( ) 3 2 30, 12 12 24

2

ii i i i

HG H l H H l H l tanh

l

= + − +

(3.2)

for smooth bearing surfaces. Detail solution of (3.1) is discussed in chapter 2. The total film

thickness iH can be defined as

i i sH h h= + for 1,2i =

Here sh is stochastic by nature and directed by the polynomial form of probability function

defined by (Christensen & Tonder 1969a, 1969b, 1970a).

The random roughness parameters , and are characterized by the stochastic part sh

and defined in chapter 2.


32

Now bring into the stochastic averaging techniques of (Christensen & Tonder 1969a,

1969b, 1970a) for transverse roughness in (3.1) and which is transformed as below.

( )12

, , , ,

i

i i

dp Vx

dx g h l

−=

(3.3)

Where,

( )

( )

3 2 2 2 2 30

2 3

, , , , 3 3 3 3 12

12 242

i i i i i i

i

i

g h l h h h h H

hl h l tan h

l

= + + + + + + + −

+ + +

(3.4)

Resorting to the magnetic fluid flow model of (Neuringer & Rosenweig, 1964) in (3.3) we

obtained

( )2

0

120.5

, , , ,i

i i

d Vxp

dx g h l

− − =

H (3.5)

where the magnitude of the magnetic field is described as

( )( )2 A L x x KL= − −H (3.6)

wherein A is a suitable constant dependent on the material to produce a field of desired

magnetic strength. The film thickness ih in the two regions is defined as below.

1ih h= for 0 x KL and (3.7)

2ih h= for KL x L (3.8)

The corresponding boundary conditions for film pressure are given by

1 2p p= at x KL= and

2 0p = at x L= (3.9)

The solution of (3.5) under the above boundary condition is given respectively, by

( )( )

( )( )2 2 2 2 2 2 2

1 0

1 1 2 2

6 60.5

, , , , , , , ,

V Vp K L x L K L

g h l g h l

= − + − + H (3.10)

( )( )2 2 2

2 0

2 2

60.5

, , , ,

Vp L x

g h l

= − + H (3.11)

3.3 Results and Discussion

33

The load bearing capacity W is obtained as

1 20

2 2KL L

KL

W b p dx b p dx = + (3.12)

which takes the form

( )( )

( )( )

0

33 1331 3 8

6 , , , , , , , ,1 1 2 2

KbL A KW K b V L

g h l g h l

−

= − + +

(3.13)

Then the load carrying capacity can be described in dimensionless form as

( )

( )( )

( )

33 32

3

1 2

13 1

488 , , , , , , ,

KKwh KW

VbL g H l g l

−− = = + +

(3.14)

Where

( )* * *3 *2 * *2 * * *2 *2 * *3 *1

*2 * 3

, , , 3 3 3 3 12

3 ( ) 3

g H l H H H H

Hl H l tanh

l

= + + + + + + + −

++ +

(3.15)

( )

( )

2* 2 * *2 *2 * *3 *, , , 1 3 3 3 3

1*2 *312 3 1 3*

g l

l l tanhl

= + + + + + + +

+ − + +

(3.16)

Where

* *1

2 2 2

3* * 0 0 2

3 322 2

2, , ,

, , ,

h lH l

h h h

H Ah

h Vh h

= = =

= = = = −


Equation (3.14) defined the dimensionless load carrying capacity. Equation (3.14) indicates

that the load bearing capacity has been improved by( )* 3 1

48

K −, as compared to the couple


34

stress fluid-based bearing system. As the expression in the (3.14) is linear concerned with

* it is easy to see that rise in the values * would lead to improved W . The magnetization

boosts the viscosity of the fluid resulting in boosted pressure and so the load carrying

capacity. The profile of W concerned with the * is displayed in Fig. 3.2 - 3.8. One can

notice from all these figures that the W increases with regards to * . However, the

influence of accompanying with roughness on the distribution of W with regards to *

remains insignificant. Further, the introduces a minimal effect on the load profile due to

magnetization. In addition, the influence of porosity on the distribution of W with regards

to * is almost insignificant up to 0.01= .

FIGURE 3.2 Profile of W

for the combination of * and K


for the combination of * and *H

0.87

1.07

1.28

1.48

1.69

1.89

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

K=0.45 K=0.55 K=0.65 K=0.75 K=0.85

1.50

1.54

1.58

1.62

1.66

1.70

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

H*=1.3 H*=1.7 H*=2.1 H*=2.5 H*=2.9


35


for the combination of * and *





1.27

1.33

1.39

1.44

1.50

1.56

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

= 0 = 0.05 = 0.10 = 0.15 = 0.

0.50

0.90

1.31

1.71

2.12

2.52

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=−0. =−0.1 =0 =0.1 = 0.

0.93

1.13

1.33

1.53

1.73

1.93

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=−0. =−0.1 =0 =0.1 = 0.


36


for the combination * and


for the combination of * and l

TABLE 3.1 Distribution of W

for the combination of * and K

001.0,30.0,10.0,10.0,05.0,10.2 ==−=−=== lH

45.0=K 55.0=K 65.0=K 75.0=K 85.0=K

0= 1.88482194 1.73985533 1.53131533 1.2476431

3

0.8772799

4 1.0= 1.88555111 1.74120950 1.53329450 1.2502473

0

0.8805091

1 15.0= 1.88591569 1.74188658 1.53428408 1.2515493

8

0.8821236

9 20.0= 1.88628027 1.74256367 1.53527366 1.2528514

7

0.8837382

7 25.0= 1.88664486 1.74324075 1.53626325 1.2541535

5

0.8853528

6

0.45

0.68

0.90

1.13

1.35

1.58

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

1.19

1.40

1.62

1.83

2.05

2.26

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


37


for the combination of * and *H

001.0,30.0,10.0,10.0,05.0,65.0 ==−=−=== lK

3.1=H 7.1=H 1.2=H 5.2=H 9.2=H

0= 1.68987292 1.56939193 1.53131533 1.51536720 1.50748966

1.0= 1.69185209 1.57137110 1.53329450 1.51734636 1.50946883

15.0= 1.69284167 1.57236068 1.53428408 1.51833595 1.51045841

20.0= 1.69383126 1.57335027 1.53527366 1.51932553 1.51144800

25.0= 1.69482084 1.57433985 1.53626325 1.52031511 1.51243758



001.0,30.0,10.0,10.0,65.0,10.2 ==−=−=== lKH

0= 05.0= 10.0= 15.0= 20.0=

0= 1.55246784 1.53131533 1.47123648 1.38109933 1.27227441

1.0= 1.55444701 1.53329450 1.47321565 1.38307850 1.27425357

15.0= 1.55543659 1.53428408 1.47420523 1.38406808 1.27524316

20.0= 1.55642617 1.53527366 1.47519481 1.38505766 1.27623274

25.0= 1.55741576 1.53626325 1.47618440 1.38604725 1.27722232

TABLE 3.4 Distibution of W




001.0,30.0,65.0,10.0,05.0,10.2 ===−=== lKH

2.0−= 1.0−= 0= 1.0= 2.0=

0 = 1.91965508 1.53131533 1.27550471 1.09421611 0.95898681

0.1 = 1.92163425 1.53329450 1.27748388 1.09619528 0.96096598

0.15 = 1.92262383 1.53428408 1.27847346 1.09718486 0.96195556

0.20 = 1.92361341 1.53527366 1.27946304 1.09817445 0.96294515

0.25 = 1.92460300 1.53626325 1.28045263 1.09916403 0.96393473

001.0,30.0,10.0,65.0,05.0,10.2 ==−==== lKH

2.0−= 1.0−= 0= 1.0= 2.0=

0= 2.50862929 1.53131533 1.02482034 0.72803698 0.53969807

1.0= 2.51060845 1.53329450 1.02679951 0.73001615 0.54167724

15.0= 2.51159804 1.53428408 1.02778909 0.73100573 0.54266682

20.0= 2.51258762 1.53527366 1.02877868 0.73199531 0.54365641

25.0= 2.51357720 1.53626325 1.02976826 0.73298490 0.54464599


38

TABLE 3.6 Distibution of W for the combination of and

2.10, 0.05, 0.10, 0.10, 0.30, 0.65H l K = = = − = − = =

0= 0001.0= 001.0= 01.0= 1.0=

0 = 1.56926302 1.56538201 1.53131533 1.25875830 0.46521341

0.1 = 1.57124218 1.56736118 1.53329450 1.26073747 0.46719258

0.15 = 1.57223177 1.56835076 1.53428408 1.26172705 0.46818216

0.20 = 1.57322135 1.56934035 1.53527366 1.26271663 0.46917175

0.25 = 1.57421093 1.57032993 1.53626325 1.26370622 0.47016133

TABLE 3.7 Distribution of W for the combination of * and l

2.10, 0.65, 0.05, 0.10, 0.10, 0.001H K = = = = − = − =

0.1l = 0.2l = 0.3l = 0.4l = 0.5l =

0 = 1.19782102 1.32232446 1.53131533 1.83593029 2.25321888

0.1 = 1.19980019 1.32430362 1.53329450 1.83790946 2.25519804

0.15 = 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

0.20 = 1.20177936 1.32628279 1.53527366 1.83988862 2.25717721

0.25 = 1.20276894 1.32727237 1.53626325 1.84087821 2.25816679

Fig. 3.9 - 3.14 depict the trends of Wwith regards to step location. These graphs underline

that the step location performs a significant role in developing the bearing characteristics. It

is remarked from Fig. 3.13 that the influence of on the load bearing capacity concerned

with step location is almost negligible, up to 0.01= . It is seen that the load bearing

capacity drops with the rise in the values of step location. Besides, bearing load reduction is

observed due to roughness.

FIGURE 3.9 Profile of W for the combination of K and *H

0.82

1.04

1.27

1.49

1.72

1.94

0.45 0.55 0.65 0.75 0.85

LO

AD

K

H*= 1.3 H*= 1.7 H*= 2.1 H*= 2.5 H*= 2.9


39

FIGURE 3.10 Profile of W for the combination of K and


for the combination of K and *



0.74

0.98

1.21

1.45

1.68

1.92

0.45 0.55 0.65 0.75 0.85

LO

AD

K

= 0 = 0.05 = 0.10 = 0.15 = 0.

0.33

0.89

1.44

2.00

2.55

3.11

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0. =−0.1 =0 =0.1 = 0.

0.56

0.92

1.29

1.65

2.02

2.38

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0. =−0.1 =0 =0.1 = 0.


40


for the combination of K and


for the combination of K and l

TABLE 3. 8 Distribution of W

for the combination of K and *H

001.0,30.0,10.0,10.0,05.0,15.0 ==−=−=== l

3.1=H 7.1=H 1.2=H 5.2=H 9.2=H

45.0=K 1.9385276

5

1.8985501

2

1.8859156

9

1.8806238

4

1.8780099

5 55.0=K 1.8379449

2

1.7649543

9

1.7418865

8

1.7322247

9

1.7274523

7 65.0=K 1.6928416

7

1.5723606

8

1.5342840

8

1.5183359

5

1.5104584

1 75.0=K 1.4951232

9

1.3100421

2

1.2515493

8

1.2270500

9

1.2149487

3 85.0=K 1.2366951

3

0.9672717

8

0.8821236

9

0.8464599

7

0.8288439

9

0.30

0.63

0.96

1.28

1.61

1.94

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=0 =0.0001 =0.001 =0.01 =0.1

0.70

1.12

1.54

1.95

2.37

2.79

0.45 0.55 0.65 0.75 0.85

LO

AD

K

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


41



0.15, 2.10, 0.10, 0.10, 0.30, 0.001H l = = = − = − = =

0= 05.0= 10.0= 15.0= 20.0= 45.0=K 1.91235103 1.88591569 1.81084143 1.69823486 1.56233524

55.0=K 1.76615553 1.74188658 1.67296167 1.56956939 1.44477262

65.0=K 1.55543659 1.53428408 1.47420523 1.38406808 1.27524316

75.0=K 1.26846267 1.25154938 1.20350363 1.13139712 1.04429835

85.0=K 0.89350224 0.88212369 0.84978839 0.80122273 0.74248968



0.15, 2.10, 0.05, 0.10, 0.30, 0.001H l = = = = − = = 2.0−= 1.0−= 0= 1.0= 2.0=

45.0=K 3.10451968 1.88591569 1.25609744 0.88816650 0.65543010

55.0=K 2.86154194 1.74188658 1.16264095 0.82388634 0.60935654

65.0=K 2.51159804 1.53428408 1.02778909 0.73100573 0.54266682

75.0=K 2.03523959 1.25154938 0.84401544 0.60432461 0.45161257

85.0=K 1.41301821 0.88212369 0.60379355 0.43864289 0.33244539



001.0,30.0,10.0,05.0,10.2,15.0 ==−==== lH

2.0−= 1.0−= 0= 1.0= 2.0=

45.0=K 2.37203501 1.88591569 1.56583959 1.33912587 1.17011196

55.0=K 2.18790829 1.74188658 1.44816458 1.24007888 1.08491942

65.0=K 1.92262383 1.53428408 1.27847346 1.09718486 0.96195556

75.0=K 1.56142569 1.25154938 1.04730875 0.90247165 0.79435288

85.0=K 1.08955792 0.88212369 0.74521296 0.64796704 0.57524386



15.0,30.0,10.0,10.0,05.0,10.2 ==−=−=== lH 0= 0001.0= 001.0= 01.0= 1.0=

45.0=K 1.94106483 1.93620750 1.89357194 1.55254834 0.56242454

55.0=K 1.79968391 1.79522695 1.75610533 1.44315863 0.53364825

65.0=K 1.59301302 1.58913201 1.55506533 1.28250830 0.48896341

75.0=K 1.30918125 1.30608370 1.27889313 1.06127726 0.42547757

85.0=K 0.93631770 0.93424303 0.91602994 0.77014538 0.34029827


42


for the combination of K and l

001.0,10.0,10.0,05.0,10.2,15.0 =−=−==== H

1.0=l 2.0=l 3.0=l 4.0=l 5.0=l

45.0=K 1.46978676 1.62512334 1.88591569 2.26612771 2.78712298

55.0=K 1.35964460 1.50233707 1.74188658 2.09109763 2.56956468

65.0=K 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

75.0=K 0.98436542 1.08412672 1.25154938 1.49550131 1.82957023

85.0=K 0.70151470 0.76897260 0.88212369 1.04687379 1.27229089

The influence of film thickness ratio *H on the variation of W is shown in Fig. 3.15-3.19.

It is perceived that initially there is a sharp decline in the load bearing capacity. Further, it is

perceived that the influence of roughness parameters and porosity on the profile of W with

regards to *H remains negligible when the film thickness ratio exceeds the value 2.5.

However, the effect of couple stress remains visibly distinct.


for the combination of *H and *



1.25

1.34

1.44

1.53

1.63

1.72

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

= 0 = 0.05 = 0.10 = 0.15 = 0.

0.52

0.96

1.41

1.85

2.30

2.74

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0. =−0.1 =0 =0.1 = 0.


43




for the combination of *H and


for the combination of *H and l

0.93

1.16

1.40

1.63

1.87

2.10

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0. =−0.1 =0 =0.1 = 0.

0.44

0.70

0.96

1.21

1.47

1.73

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=0 =0.0001 =0.001 =0.01 =0.1

1.19

1.45

1.71

1.97

2.23

2.49

1.30 1.70 2.10 2.50 2.90

LO

AD

H*

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


44



15.0,30.0,10.0,10.0,001.0,65.0 ==−=−=== lK

0 = 0.05 = 0.10 = 0.15 = 0.20 =

3.1=H 1.71517862 1.69284167 1.62930259 1.53369142 1.41776360

7.1=H 1.59368479 1.57236068 1.51177480 1.42081770 1.31089489

1.2=H 1.55543659 1.53428408 1.47420523 1.38406808 1.27524316

5.2=H 1.53944291 1.51833595 1.45839242 1.36847654 1.25995271

9.2=H 1.53154974 1.51045841 1.45056148 1.36072221 1.25230354



001.0,10.0,05.0,3.0,65.0,15.0 =−===== lK

2.0−= 1.0−= 0= 1.0= 2.0=

3.1=H 2.73367332 1.69284167 1.14478153 0.81961720 0.61123706

7.1=H 2.56070656 1.57236068 1.05781157 0.75502433 0.56212854

1.2=H 2.51159804 1.53428408 1.02778909 0.73100573 0.54266682

5.2=H 2.49213632 1.51833595 1.01458812 0.71997951 0.53338153

9.2=H 2.48285103 1.51045841 1.00785941 0.71419619 0.52838218



001.0,10.0,05.0,3.0,65.0,15.0 =−===== lK

2.0−= 1.0−= 0= 1.0= 2.0=

3.1=H 2.09563939 1.69284167 1.42454630 1.23237560 1.08758328

7.1=H 1.96229857 1.57236068 1.31505003 1.13235138 0.99579475

1.2=H 1.92262383 1.53428408 1.27847346 1.09718486 0.96195556

5.2=H 1.90633573 1.51833595 1.26285450 1.08188474 0.94696439

9.2=H 1.89836040 1.51045841 1.25507294 1.07419739 0.93936951


for the combination of *H and

10.0,10.0,05.0,3.0,65.0,15.0 −=−===== lK

0= 0001.0= 001.0= 01.0= 1.0=

3.1=H 1.73241171 1.72836718 1.69284167 1.40687457 0.54232715

7.1=H 1.61049480 1.60659509 1.57236068 1.29818760 0.49299776

1.2=H 1.57223177 1.56835076 1.53428408 1.26172705 0.46818216

5.2=H 1.55624342 1.55236644 1.51833595 1.24613397 0.45555996

9.2=H 1.54835425 1.54447843 1.51045841 1.23836001 0.44871675


45

TABLE 3. 18 Distibution of W

for the combination of *H and l

10.0,10.0,05.0,001.0,65.0,15.0 −=−===== K

1.0=l 2.0=l 3.0=l 4.0=l 5.0=l

3.1=H 1.33583581 1.46928939 1.69284167 2.01779607 2.46147831

7.1=H 1.23498265 1.36097263 1.57236068 1.88028125 2.30180008

1.2=H 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

5.2=H 1.18596219 1.31003617 1.51833595 1.82200329 2.23808679

9.2=H 1.17848977 1.30240853 1.51045841 1.81378394 2.22943426

The influence of the standard deviation shown in Fig. 3.20 - 3.23 direct that there is a

considerable decrease in W however, in the case of porosity where the effect is negligible

up to 0.01. Thus, the trio of porosity, roughness, and step location considerably influence the

performance characteristics. The flow of the lubricant is resisted by the roughness of the

bearing surfaces and which results in decreases pressure and hence the load bearing capacity.





0.49

0.91

1.32

1.74

2.15

2.57

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0. =−0.1 =0 =0.1 = 0.

0.85

1.07

1.29

1.52

1.74

1.96

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0. =−0.1 =0 =0.1 = 0.


46


for the combination of * and


for the combination of and l



10.0,10.2,3.0,001.0,65.0,15.0 −====== HlK 2.0−= 1.0−= 0= 1.0= 2.0=

0= 2.56301828 1.55543659 1.03815281 0.73668031 0.54602632

05.0= 2.51159804 1.53428408 1.02778909 0.73100573 0.54266682

10.0= 2.36917528 1.47420523 0.99793136 0.71450873 0.53283949

15.0= 2.16503865 1.38406808 0.95192902 0.68865124 0.51725202

20.0= 1.93269546 1.27524316 0.89436965 0.65552287 0.49694617

0.43

0.66

0.90

1.13

1.37

1.60

0.00 0.05 0.10 0.15 0.20

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

1.03

1.29

1.54

1.80

2.05

2.31

0.00 0.05 0.10 0.15 0.20

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


47



10.0,10.2,3.0,001.0,65.0,15.0 −====== HlK

2.0−= 1.0−= 0= 1.0= 2.0=

0= 1.95626062 1.55543659 1.29300236 1.10778294 0.97003132

05.0= 1.92262383 1.53428408 1.27847346 1.09718486 0.96195556

10.0= 1.82841755 1.47420523 1.23682210 1.06660084 0.93853497

15.0= 1.69066477 1.38406808 1.17324172 1.01933035 0.90199528

20.0= 1.52983944 1.27524316 1.09468351 0.95992792 0.85548108



10.0,01.0,10.2,3.0,65.0,15.0 −=−===== HlK

0= 0001.0= 001.0=

0.01= 1.0=

0= 1.59447472 1.59048091 1.55543659 1.27586638 0.46999293

05.0= 1.57223177 1.56835076 1.53428408 1.26172705 0.46818216

10.0= 1.50914259 1.50557272 1.47420523 1.22116778 0.46283497

15.0= 1.41472919 1.41160054 1.38406808 1.15918370 0.45419678

20.0= 1.30112373 1.29848723 1.27524316 1.08247950 0.44264572



10.0,01.0,10.2,001.0,65.0,15.0 −=−===== HK 1.0=l 2.0=l

3.0=l 4.0=l 5.0=l

0= 1.21357886 1.34095577 1.55543659 1.86955966 2.30280104

05.0= 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

10.0= 1.16402422 1.28046990 1.47420523 1.75276416 2.12717701

15.0= 1.10760911 1.21226993 1.38406808 1.62610889 1.94251534

20.0= 1.03741125 1.12837290 1.27524316 1.47717002 1.73269643

The value of W gets reduced owing to positive variance, while the value of W gets

increased with variance (-ve) (Fig. 3.24 - 3.26). Therefore, the favourable impact of

negatively skewed roughness and variance (-ve) may be considered while designing the

bearing system.


48







0.45

1.12

1.79

2.45

3.12

3.79

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=−0. =−0.1 =0 =0.1 = 0.

0.30

0.76

1.23

1.69

2.16

2.62

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

0.47

1.21

1.96

2.70

3.45

4.19

-0.20 -0.10 0.00 0.10 0.20

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


49


for the combination * and *

001.0,05.0,10.2,3.0,65.0,15.0 ====== HlK

2.0−= 1.0−= 0= 1.0= 2.0=

2.0−= 3.78166561 2.51159804 1.88554310 1.51258741 1.26496014

1.0−= 1.92262383 1.53428408 1.27847346 1.09718486 0.96195556

0= 1.18574333 1.02778909 0.90780261 0.81354250 0.73751987

1.0= 0.80621435 0.73100573 0.66900518 0.61700579 0.57276137

2.0= 0.58238079 0.54266682 0.50820315 0.47800957 0.45133568



10.0,05.0,10.2,3.0,65.0,15.0 −====== HlK 0= 0001.0= 001.0=

0.01= 1.0=

2.0−= 2.61650823 2.60561774 2.51159804 1.84889330 0.52876876

1.0−= 1.57223177 1.56835076 1.53428408 1.26172705 0.46818216

0= 1.04442639 1.04273774 1.02778909 0.89943701 0.40887004

1.0= 0.73925707 0.73842326 0.73100573 0.66451194 0.35365998

2.0= 0.54713302 0.54668296 0.54266682 0.50564173 0.30415924



10.0,05.0,10.2,001.0,65.0,15.0 −====== HK 1.0=l 2.0=l

3.0=l 4.0=l 5.0=l

2.0−= 1.81730027 2.07097540 2.51159804 3.18713119 4.18108805

1.0−= 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

0= 0.84579738 0.91446259 1.02778909 1.18911650 1.40337395

1.0= 0.62305103 0.66402269 0.73100573 0.82514291 0.94819377

2.0= 0.47466729 0.50056658 0.54266682 0.60137307 0.67740997

From Fig. 3.29 and some of the earlier graphs one can easily accomplish that the porosity

effect is at the best nominal. However, the positive effect of couple stress is manifest as can

be seen from Fig 3.19, 3.23, 3.26, 3.28, 3.29. Indeed, due to the existence of microstructure

additive in the lubricant gives rise to an increase in film pressure and therefore the load

bearing capacity. The couple stress parameter delivers a mechanism for the interface of the

lubricant with bearing geometry.


50







0.39

0.71

1.03

1.35

1.67

1.99

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

0.82

1.30

1.78

2.27

2.75

3.23

-0.20 -0.10 0.00 0.10 0.20

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

0.43

0.81

1.20

1.58

1.97

2.35

0.00 0.03 0.05 0.08 0.10

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


51

Some of the graphical representations make it sure, that the positive influence of magnetic

fluid lubrication under the couple stress effect could be enough to overcome the adversarial

effect caused due to roughness and porosity, for a suitable choice of step location. Nowadays

ferrofluid is available easily and therefore ferrofluid lubrication of this type of bearing

system will be favourable to the industry for expanding the life span of the bearing system.

TABLE 3. 26 Distribution of W


3.0,10.0,05.0,10.2,65.0,15.0 =−===== lHK 0= 0001.0= 001.0=

01.0= 1.0=

2.0−= 1.98318717 1.97695637 1.92262383 1.51000971 0.49610482

1.0−= 1.57223177 1.56835076 1.53428408 1.26172705 0.46818216

0= 1.30446746 1.30181930 1.27847346 1.08493549 0.44337530

1.0= 1.11610315 1.11418112 1.09718486 0.95260543 0.42118633

2.0= 0.97634221 0.97488355 0.96195556 0.84980964 0.40121823



10.0,05.0,10.2,001.0,65.0,15.0 −====== HK 1.0=l 2.0=l

3.0=l 4.0=l 5.0=l

2.0−= 1.42327722 1.60322231 1.92262383 2.43166182 3.22817785

1.0−= 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

0= 1.03967134 1.13098094 1.27847346 1.48137779 1.73834419

1.0= 0.91757574 0.98744494 1.09718486 1.24217230 1.41652209

2.0= 0.82183292 0.87705289 0.96195556 1.07083585 1.19705836



10.0,05.0,10.2,10.0,65.0,15.0 −===−=== HK 1.0=l 2.0=l

3.0=l 4.0=l 5.0=l

0= 1.22365485 1.35333364 1.57223177 1.89405700 2.34034362

0001.0= 1.22132814 1.35047493 1.56835076 1.88838969 2.33164154

001.0= 1.20078977 1.32529321 1.53428408 1.83889904 2.25618763

01.0= 1.02867955 1.11792985 1.26172705 1.45880650 1.70717236

1.0= 0.43365982 0.44791627 0.46818216 0.49164169 0.51588289

The comparative percentage growth in W for various values of H and l is displayed in

Table 3.29. It is noticed an improvement of nearly 86% for 2.9H = and 0.5l = compared

to conventional lubricant based bearing structure.


52


and W

R for various values of H and l

H l

W with traditional

lubrication

( )0, 0.65K = =

W of the present study with

0.15, 0.05, 0.10, 0.10,

0.001, 0.65K

= = = − = − = =

%

increase

in W

R

1.3

0.1 0.8725859664

1.0371639445

1.3358358147

1.6928416729

53.09

0.3 1.0371639445

1.6928416729

63.22

0.5 1.3586096101

2.4614783060

81.18

2.1

0.1 0.7753508963

1.2007897734

54.87

0.3 0.9259439072

1.5342840808

65.70

0.5 1.2202714077

2.2561876265

84.89

2.9

0.1 0.7568027125

1.1784897679

55.72

0.3 0.9062415227

1.5104584143

66.67

0.5 1.1983598110

2.2294342577

86.04

The comparative percentage growth in W for distinct values of K and l is presented in

Table 3.30. It is a noticeable improvement of nearly 86% for 0.45K = and 0.5l =

compared to traditional lubricant based bearing structure.


and W

R for various values of K and l

K l

W with traditional

lubrication

( )0, 2.10H = =

W of the present study with

0.15, 0.05, 0.10, 0.10,

0.001, 2.10H

= = = − = − = =

%

increase

in W

R

0.45

0.1 0.9439993955

1.4697867588

55.70

0.3 1.1313535585

1.8859156909

66.70

0.5 1.4975836973

2.7871229752

86.11

0.65

0.1 0.7753508963

1.2007897734

54.87

0.3 0.9259439072

1.5342840808

65.70

0.5 1.2202714077

2.2561876265

84.89

0.85

0.1 0.4633281963

0.7015146968

51.41

0.3 0.5459080675

0.8821236906

61.59

0.5 0.7072058909

1.2722908915

79.90

3.4 Conclusion

The graphical depiction shows that the adversarial influence of roughness can be

remunerated up to a significant level by the affirmative influence of couple stress and

3.4 Conclusion

53

magnetization in the case of negatively skewed roughness particularly when variance (-ve)

occurs.

This study directs that the roughness features need to be given attention while

designing the bearing system even if an appropriate combination of magnetic parameter and

couple stress parameter is in place.

Needless to say, is that the favorable effect of a couple stress enriches by the

ferrofluid lubrication.From Tabular results it is observed that load capacity is increased by

86% compared to non magnetic case.

A distinct feature of this type of bearing system is that in spite of the presence of a

half a dozen of parameters making down the load; the bearing supports some extent of load

even in the lack of flow, which does not occur in the situation of traditional lubricant based

bearing system.

Equation (3.1) is modified in the next chapter to examine the performance of

longitudinalsurface roughness with couple stress.

54

CHAPTER 4

4. Performance of Ferrofluid Based

Longitudinally Rough Porous Parallel

Stepped Plates with Couple Stress

4.1 Introduction

Nowadays the flow of non-Newtonian fluids has been extensively used in many industries

and current technology which, directed numerous researchers to attempt various flow

problems associated with non-Newtonian fluids. One of the most important theory of couple

stress fluid given by (Stokes,1966). To overcome the necessity of a recent machine system

working severe situations, the enlarged use of different types of non-Newtonian fluid as

lubricant has been highlighted. The use of additives in the lubricant reduces the compassion

of lubricants to changes in shear rate and which supports restored load bearing capacity and

response time. The micropolar theory for porous parallel stepped plates analyzed by

(Siddangouda,2015b) concluded that the impact of non-Newtonian micropolar fluid initiate

to improve the load bearing capacity.

Several studies have been made on the hydrodynamic squeeze film lubrication with

couple stress fluid and the studies discovered that the couple stress fluid boosted the load

carrying capacity and response time as related to the Newtonian case(Biradar , 2012; Biradar,

2013; Lin, 1998; Ramanaiah & Sarkar, 1978; Naduvinamani & Siddangouda, 2009; Lin et

al.,2006). Lubrication plays a significant role in bearing as it reduces the friction in the

bearing. Nowadays ferrofluid as a lubricant is used in various engineering applications like

material science, heat transfer, dynamic sealing, damping, etc. It is a liquid that becomes

strongly magnetized in the existence of the magnetic field. The various investigator has used

55

ferrofluid as a lubricant for their study. Shah (2003) studied the influence of ferrofluid on

step bearing by two steps. His study revealed that the overall performance of the bearing

upgraded by ferrofluid. In the current years, surface roughness and its impact on machine

design are vital features that have been widely studied. Some models have been proposed to

study the consequence of surface roughness on the bearing performance. Due to the random

structure of the surface roughness, a stochastic model for the study of hydrodynamic

lubrication has been developed by (Christensen & Tonder, 1969a, 1969b, 1970a) and this

model has been used by many researchers for example (Andharia & Deheri, 2004) analyzed

the influence of longitudinal roughness with ferrofluid based squeeze film lubrication in

truncated conical plates. The study concluded that pressure, load carrying capacity and

response time enhanced due to ferrofluid lubrication. Shimpi and Deheri (2014) extended

the work of (Andharia & Deheri, 2011) by considering the deformation effect with slip

velocity. Patel and Deheri (2016b) investigated the ferrofluid lubrication on longitudinally

rough conical plates with slip velocity. This investigation proposed that the negative

influence of slip and roughness can be compensated with the positive influence of

magnetization and standard deviation. Patel et al. (2017b) extended the above work with

considering deformation effect and different form of magnitude of the magnetic field.

Ramesh et al. (2013) studied numerically the rough porous rectangular plates with the

magnetic field. It was observed from their study pressure and load bearing capacity are found

to be enhanced as increasing the values of Hartmann number and roughness parameter.

Vashi et al. (2018) investigated the combined influence of surface roughness and ferrofluid

lubrication on parallel stepped plates in the existence of couple stress fluid. Their study

discovered that load bearing capacity is enhanced by use of ferrofluid as a lubricant.

So, this chapter has been made to study the performance characteristic of longitudinally

rough porous parallel stepped plates with ferrofluid based couple stress fluid.

4.2 Analysis

Fig. 4.1 displays the physical structure of the bearing system. The upper plate is moving

towards the fixed lower porous plate with normal velocity V . The film region is filled by

incompressible ferrofluid based couple stress fluid.

Performance of Ferrofluid Based Longitudinally Rough Porous Parallel Stepped Plates

with Couple Stress

56

FIGURE 4.1 Configuration of longitudinally rough porous parallel stepped plates

Using the traditions of hydrodynamic lubrication, the generalized Reynolds’ type equation

leading the pressure distribution turns out to be (Biradar, 2012).

( )12

,

i

i i

dp Vx

dx G H l

−=

(4.1)

where,

( ) 3 2 30, 12 12 24

2

ii i i i

HG H l H H l H l tanh

l

= + − +

(4.2)

for smooth bearing.

Now in the view of stochastic averaging techniques of (Christensen & Tonder, 19969a,

1969b, 1970a) for longitudinal roughness with a nonzero mean (4.1) reduce to the form given

below.

( )12

,

i

i i

dp Vx

dx g H l

−=

(4.3)

Where,

( )( ) ( )

( )1

2 303 1

1

1 1, 12 12 24

2

i

i i

i i

E Hg H l H l l tanh

lE H E H

−

− −

= + − +

(4.4)

4.2 Analysis

57

( )( ) ( )

( )2 30

1

, , ,1 1, , , , 12 12 24

, , , , , , 2

ii i

i i i i

n hg h l H l l tanh

m h n h l

= + − +

(4.5)

( ) ( ) ( )( )3 1 2 2 2 3 2 3, , , 1 3 6 10 3i i i i i im h h h h h− − − −= − + + − + + (4.6)

( ) ( ) ( )( )1 1 2 2 2 3 2 3, , , 1 3i i i i i in h h h h h− − − −= − + + − + + (4.7)

Where,

1ih h= for 0 x KL and (4.8)

2ih h= for KL x L (4.9)

Resorting to the magnetic fluid flow model of (Neuringer & Rosenweig 1964) (4.3) transfer

to

( )2

0

120.5

,i

i i

d Vxp

dx g h l

− − =

H (4.10)

where

( )( )2 A L x x KL= − −H (4.11)

wherein A is a suitable constant dependent on the material to produce a field of desired

magnetic strength.

The pressure boundary conditions are

1 2p p= at x KL= and 2 0p = at x L= (4.12)

The solution of (4.10) under the above boundary condition is given respectively, by

( )( )

( )( )2 2 2 2 2 2 2

1 0

1 1 2 2

6 60.5

, , , , , , , ,

V Vp K L x L K L

g h l g h l

= − + − + H

(4.13)

( )( )2 2 2

2 0

2 2

60.5

, , , ,

Vp L x

g h l

= − + H (4.14)

Where,


with Couple Stress

58

( )( ) ( )

( )

21 1 0

1 1 1 1

13

1 1, , , , 12 12

, , , , , ,

1

, , ,24

2

g h l H lm h n h

n hl tanh

l

= + − +

(4.15)

( )( ) ( )

( )22 32 2 0

2 2 2

1

, , ,1 1, , , , 12 12 24

, , , , , , 2

n hg h l H l l tanh

m h n h l

= + − +

(4.16)

The expression of load capacity is achieved as

1 20

2 2KL L

KL

W b p dx b p dx = + (4.17)

which takes the form

( )( )

( )( )

33 1331 3 8

6 , , , , , , , ,1 1 2 2

KbL Aμ K0W K b VLg h l g h l

−

= − + +

(4.18)

Using the following dimensionless terms

3* * * * * *0 2 01

3 32 2 2 2 2 2

2- , , , , , ,

Ah Hh lH l

V h h h h h h= = = = = = =

in the (4.18) the expression for dimensionless load capacity can be described as

( ) ( )33 32

31 2

13 1

488

KKWh KW

G GVbL

−− = = + +

(4.19)

Where

( )

( )

232

1

1 1 1 1

3

1

1 312

( , , , , ) ( , , , ) , , ,

13

, , ,

h lG

g h l M H N H

l tanhN H l

= = + − +

(4.20)

4.3 Result and Discussion

59

( ) ( )

( )

232

2

2 2 2 2

3

2

1 312

, , , , ( , , ) , ,

13

, ,

h lG

g h l M N

l tanhN l

= = + − +

(4.21)

Where

( )( )

( )

1 2 2 2

31

3 2 3

1 3 6, , ,

10 3

H HM H H

H

− −

−

−

− + + − = + +

(4.22)

( ) ( ) ( )( )1 1 2 2 31 , , , 1 32 2 3N H H H H H − − − − = − + + − + +

( ) ( ) ( )( )2 2 2 32 , , 1 3 6 10 3M = − + + − + +

( ) ( ) ( )2 2 2 3

2 , , 1 3N = − + + − + +


This study purposes to examine the influence of a ferrofluid based squeeze film between

longitudinally rough stepped plates considering couple stress effect. Equation (4.19) presents

the dimensionless load carrying capacity. The load carrying capacity is enhanced by using

ferrofluid as a lubricant as related to the couple stress fluid-based bearing system. Also, the

comparison is made between the ferrofluid based bearing system and couple stress fluid-

based bearing system. Fig. 4.2 - 4.8 display the distribution of W with regards to for

various values of , , , l . It is perceived that the W rises as the values of

increases. Fig 4.8 describes that influence of can found noticeable for different values of

l .


with Couple Stress

60

FIGURE 4.2 Profile of W for the combination of * and K

FIGURE 4.3 Profile of W for the combination of * and *H

FIGURE 4.4 Profile of W for the combination of *μ and *

0.74

0.91

1.08

1.26

1.43

1.60

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

K=0.45 K=0.55 K=0.65 K=0.75 K=0.85

1.27

1.31

1.34

1.38

1.41

1.45

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

H*=1.3 H*=1.7 H*=2.1 H*=2.5 H*=2.9

1.27

1.34

1.41

1.47

1.54

1.61

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

= 0 = 0.05 = 0.10 = 0.15 = 0.


61

FIGURE 4.5 Profile of W for the combination of * and *

FIGURE 4.6 Profile of W for the combination of * and *

FIGURE 4.7 Profile of W for the combination of * and

1.06

1.13

1.20

1.26

1.33

1.40

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=−0.05 =−0.05 =0

=0.05 = 0.05

0.50

0.72

0.95

1.17

1.40

1.62

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=−0.05 =−0.05 =0 =0.05 = 0.05

0.44

0.62

0.80

0.98

1.16

1.34

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1


with Couple Stress

62

FIGURE 4.8 Profile of W for the combination of * and l

The influence of step location on W with regards to * * * *, , , , ,H l is presented in Fig

4.9 - 4.14. As the values of K rises, the value of W is falls down.

FIGURE 4.9 Profile of W for the combination of K and H


1.00

1.19

1.37

1.56

1.74

1.93

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

0.70

0.89

1.08

1.27

1.46

1.65

0.45 0.55 0.65 0.75 0.85

LO

AD

K

H*= 1.3 H*= 1.7 H*= 2.1 H*= 2.5 H*= 2.9

0.96

1.29

1.62

1.94

2.27

2.60

0.45 0.55 0.65 0.75 0.85

LO

AD

K

= 0 = 0.05 = 0.10 = 0.15 = 0.


63




0.32

0.66

1.00

1.34

1.68

2.02

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0.05 =−0.05 =0 =0.05 = 0.05

0.62

0.84

1.06

1.28

1.50

1.72

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0.05 =−0.05 =0

=0.05 = 0.05

0.28

0.55

0.82

1.10

1.37

1.64

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=0 =0.0001 =0.001 =0.01 =0.1


with Couple Stress

64

FIGURE 4.14 Profile of W for the combination of K and l

The influence of H on W with regards to * * *, , , , l is presented in Fig. 4.15 - 4.18.

Influence H on load bearing capacity can be seen adversely. From Fig 4.16 it is perceived

that the load drop is nominal with regards l . From Fig 4.17 noticed that the initial influence

of on W is insignificant up to 0.001= .

FIGURE 4.15 Profile of W for the combination of H and

0.60

0.96

1.31

1.67

2.02

2.38

0.45 0.55 0.65 0.75 0.85

LO

AD

K

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

1.26

1.36

1.46

1.56

1.66

1.76

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

= 0 = 0.05 = 0.10 = 0.15 = 0.


65




0.48

0.74

1.00

1.27

1.53

1.79

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0.05 =−0.05 =0 =0.05 = 0.05

1.05

1.15

1.25

1.35

1.45

1.55

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0.05 =−0.05 =0

=0.05 = 0.05

0.40

0.61

0.83

1.04

1.26

1.47

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=0 =0.0001 =0.001 =0.01 =0.1


with Couple Stress

66

FIGURE 4.19 Profile of W for the combination of H and l

Fig 4.20 - 4.22 presents the impact on load bearing capacity for various values of

* *, , , l . Increasing the value of the value of W is also rises.

FIGURE 4.20 Profile of W for the combination of and


1.00

1.22

1.44

1.66

1.88

2.10

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

1.03

1.17

1.31

1.46

1.60

1.74

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0.05 =−0.05 =0

=0.05 = 0.05

0.48

0.78

1.07

1.37

1.66

1.96

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0.05 =−0.05 =0 =0.05 = 0.5


67


FIGURE 4.23 Profile of W for the combination of and l

The distribution of on W with respects , ,l can be presented in Fig. 4.24 - 4.26.


0.43

0.67

0.92

1.16

1.41

1.65

0.00 0.05 0.10 0.15 0.20

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

1.01

1.32

1.63

1.93

2.24

2.55

0.00 0.05 0.10 0.15 0.20

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

0.32

0.60

0.88

1.17

1.45

1.73

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

=−0.05 =−0.05 =0 =0.05 = 0.05


with Couple Stress

68



The impact of on W with regards to l and can be described in Fig. 4.27 - 4.28.

Negatively increases the value of increases the value of W while the W is decreasing

as the values of growing positively.

0.40

0.61

0.81

1.02

1.22

1.43

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

0.86

1.10

1.35

1.59

1.84

2.08

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


69



Table 4.1 and 4.2 characterize the comparison of the current study with the traditional

lubricant case. The relative increase in W (W

R ) is calculated in Table 4.1 and Table 4.2.

It is noticed from Table 4.1 that the comparative percentage of growth in W associated

with traditional lubrication is around 58 % when 2.9H = and 0.5l =

0.46

0.92

1.37

1.83

2.28

2.74

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

0.29

0.57

0.84

1.12

1.39

1.67

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1


with Couple Stress

70


R for distinct values of H and l

H

l

W with

traditional

lubrication

( )0, 0.65K = =

W for the current study

with

0.15, 0.15, 0.025,

0.025, 0.001, 2.10H

= = = − = − = =

% increase in W

(W

R )

1.3

0.1 0.8725859664

1.147819798

31.54

0.3 1.0371639445

1.439048284

38.75

0.5 1.3586096101

2.096916842

54.34

2.1

0.1 0.7753508963

1.029975141

32.84

0.3 0.9259439072

1.301994245

40.61

0.5 1.2202714077

1.921507158

57.47

2.9

0.1 0.7568027125

1.009890579

33.44

0.3 0.9062415227

1.280603615

41.31

0.5 1.1983598110

1.897608187

58.35

From Table 4.2 It is noticed that the comparative percentage of growth in W associated

with traditional lubrication is around 58 % when 0.65K = and 0.5l = .


R for distinct values of K and l

K l

W with

traditional

lubrication

( )0, 0.65K = =

W of the current study with

0.15, 0.15, 0.025,

0.025, 0.001, 2.10H

= = = − = − = =

% increase in

W

(W

R )

0.45

0.1 0.9439993955

1.2586007714

33.33

0.3 1.1313535585

1.5979407000

41.24

0.5 1.4975836973

2.3713091919

58.34

0.65

0.1 0.7753508963

1.0299751412

32.84

0.3 0.9259439072

1.3019942454

40.61

0.5 1.2202714077

1.9215071582

57.47

0.85

0.1 0.4633281963

0.6053925722

30.66

0.3 0.5459080675

0.7528589792

37.91

0.5 0.7072058909

1.0877181095

53.81

4.4 Conclusion

71

4.4 Conclusion

The squeeze film based ferrofluid lubrication for longitudinally rough porous parallel

stepped plates with couple stress effect is studied. Based on the graphical and tabular results

following conclusions are made.

The influence of ferrofluid lubrication combined with a couple stress effect enhances

the load bearing capacity compared to the traditional lubricant based bearing structure.

The load increases almost 58% greater associated with the traditional lubricant based

bearing system.

From the industry point of view, the longitudinal roughness turns out to be more

favourable as compared to transverse roughness.

The contrary influence cause due to porosity and roughness can be compensated with

the proper selection of step location with magnetization parameter and couple stress

parameter.

72

CHAPTER 5

5. Influence of Ferrofluid Based Doubled

Layered Porous Conical Bearing with two

Different Forms of Transverse Roughness

5.1 Introduction

This chapter aims to determine the enactment of double layered porous rough conical plates

with ferrofluid based squeeze film lubrication. The Neuringer – Roseinweig model has been

employed for magnetic fluid flow. For the characterization of roughness two different forms

of polynomial distribution function have been used and comparison is made between both

roughness structure. The stochastic model of Christensen and Tonder regarding transverse

roughness has been invoked to develop the associated Reynolds’ equation from which the

pressure circulation is found. This provides growth to the calculation of load bearing

capacity. The results presented here confirm that the introduction of double layered plates

results in improved load carrying capacity. This is further enhanced by the ferrofluid

lubrication.

Porous bearing is used very widely in many devices such as vacuum cleaners, extractor fans,

motorcar starters, hairdryer, etc. They are also used in business machines, farm and

construction equipment, and aircraft automotive accessories. In addition, the porous bearing

can work hydrodynamically longer without maintenance and more stable than the equivalent

conventional bearing. Also, in these bearings’ friction is less as compared to the non-porous

bearings. So many researchers have studied the effect of double layered porous bearings of

various shapes. Uma Srinivasan (1977b) worked to study double layered slider bearing with

a porous surface. The double layered surface enhanced the bearing’s load carrying capacity

5.1 Introduction

73

as well as the friction drag. However, it reduced the friction coefficient. Verma (1983)

investigated the influence of a doubled layered porous slider bearing. The study of (Rao et

al., 2013) focused on the relation between the Brinkman model and a double layered porous

journal bearing’s performance. The results suggested that in a double layered bearing, the

low permeability layer stuck to the high permeability one, leading to increased bearing

capacity and as a result, a decreased friction coefficient. Prakash and VIJ (1973a) analyzed

the effect of the shape of the plate and porosity on the performance of squeeze films between

porous plates of various shapes.

Uma Srinivasan (1977a) intended to study the impact of time-height of squeeze films on a

bearing’s load capacity. Various geometrical aspects like circular, elliptical, rectangular, etc.

were used for the purpose. It was a comparative study focusing on two-layered porous

bearing and conventional bearings. The results suggested that double layered plates enhance

a bearing’s load carrying capacity. Cusano (1972) analyzed an infinitely long two layered

porous bearing.

Conical bearings have been developed for use in agricultural and construction

machinery, for the suspension of jolts and insulation of engine vibrations from cabins. Lin

et al. (2012) studied the behavior of non-Newtonian micropolar fluid squeeze film between

conical plates. The non-Newtonian effects of micropolar fluid were found to be better in

comparison with the Newtonian case also its effect lengthened the approaching time of

squeeze film conical plates. Dinesh Kumar et. al. (1992) studied the effect of ferrofluid

squeeze film for spherical and conical bearings using perturbation analysis.

Practically, a perfectly smooth surface does not exist as all surfaces are rough to some

extent. In applied settings, a smooth surface bearing does not provide an optimum idea of

performance and bearing life span. Thus, in the recent year studies have focused on

correlating surface roughness with the bearing capacity. Christensen & Tonder (1969a,

1969b, 1970a) worked on the stochastic surface roughness theory with hydrodynamic

lubrication. Many authors have used this technique to understand the impact of surface

roughness on performance. Patel & Deheri (2014a) made a comparative study of different

porous configurations and their impact on double layered slider bearing with roughness and

magnetic fluid. The results suggested that the Kozeny Carman model is more effective than

Irmay’s model. Deheri et al. (2013) made a theoretical study of the influence of squeeze film

with a magnetic fluid on porous rough conical plates. The results showed that an appropriate

semi vertical angle can revert the negative impacts of porosity and standard deviations for

Influence of Ferrofluid Based Doubled Layered Porous Conical Bearing with two Different

Forms of Transverse Roughness

74

negatively skewed roughness. Patel & Deheri (2016b) deliberated the impact of slip velocity

on a squeeze film with ferrofluid in conical plates with longitudinal roughness. It was found

that standard deviation and magnetization can substantially neutralize the negative impact

created by slip velocity and surface roughness on bearing performance, provided that the

negatively skewed roughness was appropriate. Vashi et al. (2018) studied the impact of

ferrofluid based rough porous parallel plates with couple stress effect.

Various good research articles are available in the literature for the study of squeeze

film lubrication of conical bearings and truncated conical bearings. For examples (Shimpi &

Deheri, 2014) in porous truncated conical plates, (Patel & Deheri, 2007) in porous conical

plates, (Vadher et al., 2010) in porous rough conical plates (Patel et al., 2017b) in rough

conical bearing with deformation effect.

At present no work has been made to study the influence of surface roughness with

two different patterns on ferrofluid based squeeze film in double layered porous conical

plates. So, in this current chapter, the investigation of (Patel & Deheri , 2013) is extended to

the double layered porous conical plates with two different forms of the transverse surface

roughness patterns.

5.2 Analysis

All the traditions of hydrodynamic lubrication are considered here. The lubricant is

incompressible ferrofluid lubrication, considered for the analysis. Both the porous facings

are supposed to be homogeneous and isotropic and porosity is directed by a generalized form

of Darcy’s law.

Fig. 5.1 display the geometrical structure of squeeze film lubrication of porous rough

conical plates bearing.

5.2 Analysis

75

FIGURE 5.1. Configuration of rough conical bearing

In the sought of the discussion of (Uma Srinivasan, 1977a) the modified Reynolds equation

comes out to be

•

0

3 31 1 2 2

121

12 12

h sinωd dp

x dx dx H sin H H

=

+ +

(5.1)

The bearing faces are deliberated to be transversely rough in the context of (Christensen &

Tonder, 1969a, 1969b, 1970a) the lubricant total film thickness is taken as

sH = h +h (5.2)

where h is mean film thickness and sh is the part due to the surface roughness as measured

from nominal film thickness. According to (Christensen & Tonder, 1969a, 1969b, 1970a )

stochastic part sh is defined by the polynomial probability distribution function 1)( shf for

the domain chc s − , where c represents the maximum deviation from the mean film

thickness.

( )( )

32 2

7

1

35,

32

0 , elsewhere

s ss

c h c h cf h c

− −

=

(5.3)

Further, a different form of this type of polynomial distribution from (Prajapati, 1995) is



76

( )( )

22 2

5

2

15,

16

0 , elsewhere

s ss

c h c h cf h c

− −

=

(5.4)

The measure of the symmetry of the random variable sh is mean the standard deviation

and the parameter defined by the relations

( )sE h= ( )22

sE h = −

( )3

sE h = −

(5.5)

where ( )•E is the expectancy operator given by the formula

( ) ( ) ( )s sE f h dh

−• = • (5.6)

The detailed study regarding the roughness model can be observed from (Christensen &

Tonder, 1969a, 1969b, 1970a).

Neuringer and Rosensweig (1964) established a model to designate the stable flow of

magnetic fluid. This model involves the following equations.

Equation of motion:

( ) ( )20. .q q p q M H = − + + (5.7)

Equation of continuity:

. 0q = (5.8)

Maxwell’s equations:

0= H (5.9)

( ) 0. =+ MH (5.10)

Equation of magnetization:

M H= (5.11)

The magnetic field deliberated here is slanting to the lower surface and its magnitude is

defined as

( )2 2 2 2A a x sin= −H

5.2 Analysis

77

wherein A is a suitably chosen constant depending on the material to produce a field of

desired magnetic strength.

Now using the averaging procedures of (Christensen & Tonder, 1969a, 1969b, 1970a) and

(Neuringer & Rosensweig, 1964) model of magnetic fluid flow (5.1) transfer to the form

( ) 00

1

120.5 21 d d

d d

h sinx p

x x x g

•

− =

H (5.12)

wherein

( ) 3 3 2 2 2 2 2 31

1 1 2 2

3 3 3 3

12 12

g h h sin h sin h sinω h sin

H H

= + + + + + + +

+

(5.13)

For the different form of the polynomial probability distribution function, (5.1) transfer to

the form

( )( )

•

2 00

2

1210.5

h sinωd dx p

x dx dx g h

− =

H (5.14)

Where

( ) 3 3 2 2 2 2 2 32

1 1 2 2

4 3 2 4

12 12

g h h sin h sin hsin h sin

H H

= + + + + + + +

+

(5.15)

The related pressure boundary conditions are

( )0

0 0x

dpp acosecω and

dx =

= =

(5.16)

Solving expressions (5.12) and (5.14) with the suitable boundary conditions (5.16) the

appearance for dimensional less pressure established in the film region is found as

( ) ( )2 2

1

1 3 1

2

X sinω XP

G

− −

= +

(5.17)

wherein



78

( )1 3 *2 *2 * 2 *2 * *3 *1 3

0

1 2

3 3 3 3

12 12

g hG sin sin sin sin

h= = + + + + + + +

+

(5.18)

30 0

30 0

0

1 1

1 30 0

302 2

2 320

0

, , ,

, ,

, ,

h K

h hh

H

h h

h cosecω pH xX P

a cosecωhh a

•

•

−= = =

= =

−= = =

the expression for dimensional less pressure for a different form of transverse roughness is

found as

( ) ( )2 2

2

1 3 1

2

X sinω XP

G

− −

= +

(5.19)

wherein

( )2 3 *2 *2 * 2 *2 * *3 *2 1 23

0

4 3 2 4 12 12g h

G sin sin sinh

= = + + + + + + + + (5.20)

Now the load bearing capacity of the bearing can be achieved integrating the film pressure

over the squeezing film region as following

( )cos

0

2a ec

W x p x dx=

(5.21)

Lastly, the expression for W

is given by

30

04 2

h WW

h a cosec ω

•= −

= 1

3

2

cosecω

G 4

+

(5.22)

for the different form of roughness structure expression for W is obtained as

2

3

2

cosecW

G 4

= +

(5.23)

5.3 Results and discussion

79


The influence of ferrofluid squeeze film in the double layered porous rough conical plate is

studied with different forms of transverse surface roughness, also the comparison is made

between two different structures of transverse surface roughness. In the nonexistence of

magnetization, this investigation reduces the work of (Uma Srinivasan, 1977a) for smooth

bearing. The analytic expression for W and P for both the roughness structure is given by

the (5.17), (5.19), (5.22) and (5.23). It is found that W increases

4

times as related to the

traditional lubrication-based bearing system. Here the solid line represents the roughness

structure 1G and the dotted line represents the roughness structure 2G . Figures 5.2 - 5.7

describe the profile of W with regards to for different values of 1, , , and 2

It is detected that W increases as growing the values of for both the roughness structures.

It is clearly seen from Fig. 5.4 and 5.5 that influence of 1G is more related to 2G .



0.61

1.08

1.54

2.01

2.47

2.94

0.10 0.58 1.05 1.53 2.00

LO

AD

=0.1

=0.

=0.3

=0.4

=0.5

=0.1

=0.

=0.3

=0.4

=0.5



80



*




for the combination of * and 1

0.40

0.80

1.20

1.60

2.00

2.40

0.10 0.58 1.05 1.53 2.00

LO

AD

=−0.0

=−0.10

=0

=0.10

=0.0

=−0.0

=−0.10

=0

=0.10

=0.0

0.50

0.86

1.23

1.59

1.96

2.32

0.10 0.58 1.05 1.53 2.00

LO

AD

=−0.

=−0.1

=0

=0.1

=0.

=−0.

=−0.1

=0

=0.1

=0.

*

0.52

0.87

1.21

1.56

1.90

2.25

0.10 0.58 1.05 1.53 2.00

LO

AD

1=0

1=0.005

1=0.01

1=0.015

1=0.0

1=0

1=0.005

1=0.01

1=0.015


81





The advarse influence of on W

with regards to different values of 1, , , 2 and

can be seen from Fig. 5.8 - 5.12. From Fig. 5.11 it is noticed that the adverse influence of

is registered to be nominal for different values of 2ψ .

0.12

0.59

1.06

1.52

1.99

2.46

0.10 0.58 1.05 1.53 2.00

LO

AD

=0

=0.00019

=0.00190

=0.019

=0.19

=0

=0.00019

=0.0019

=0.019

=0.19

2

0.46

0.90

1.34

1.77

2.21

2.65

0.10 0.58 1.05 1.53 2.00

LO

AD

=40

=45

=50

=55

=0

=40

=45

=50

=55

=0



82

.


for the combination of and




for the combination of and 1

0.59

0.64

0.68

0.73

0.77

0.82

0.10 0.20 0.30 0.40 0.50

LO

AD

=−0.

=−0.1

=0

=0.1

= 0.

=−0.

=−0.1

=0

=0.1

= 0.

1.02

1.06

1.09

1.13

1.16

1.20

0.10 0.20 0.30 0.40 0.50

LO

AD

=−0.

=−0.1

=0

=0.1

= 0.

=−0.

=−0.1

=0

=0.1

1.03

1.07

1.11

1.15

1.19

1.23

0.10 0.20 0.30 0.40 0.50

LO

AD

1=0

1=0.005

1=0.01

1=0.015

1=0.0

1=0

1=0.005

1=0.01

1=0.015

1=0.0


83





The Fig. 5.13 - 5.18 present the profile of W with regards to roughness parameter and

for different values of 1 2, , . It is perceived that the negatively rising values of

and rises the load bearing capacity while the positively rising values of

and

decrease the values of W. The negatively skewed roughness in conjunction with variance

(-ve) may offer necessary help for the ferrofluid lubrication of the bearing system.

0.80

0.89

0.97

1.06

1.14

1.23

0.10 0.20 0.30 0.40 0.50

LO

AD

=0

=0.00019

=0.00190

=0.019

=0.19

=0

=0.00019

=0.0019

=0.019

0.99

1.05

1.11

1.16

1.22

1.28

0.10 0.20 0.30 0.40 0.50

LO

AD

=40

=45

=50

=55

=0

=40

=45

=50

=55

=0



84





FIGURE 5.15. Profile of W


1.09

1.22

1.36

1.49

1.63

-0.20 -0.10 0.00 0.10 0.20

LO

AD

1=0

1=0.005

1=0.01

1=0.015

1=0.0

1=0

1=0.005

1=0.01

1=0.015

1=0.0

0.80

1.04

1.27

1.51

1.74

1.98

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=0

=0.00019

=0.00190

=0.019

=0.19

=0

=0.00019

=0.0019

=0.019

0.98

1.10

1.21

1.33

1.44

1.56

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=40

=45

=50

=55

=0

=40

=45

=50

=55

=0


85







1.15

1.23

1.31

1.40

1.48

1.56

-0.20 -0.10 0.00 0.10 0.20

LO

AD

1=0

1=0.005

1=0.01

1=0.015

1=0.0

1=0

1=0.005

1=0.01

1=0.015

1=0.0

0.82

1.02

1.22

1.42

1.62

1.82

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=0

=0.00019

=0.00190

=0.019

=0.19

=0

=0.00019

=0.00190

=0.019

1.13

1.43

1.74

2.04

2.35

2.65

-0.20 -0.10 0.00 0.10 0.20

LO

AD

=40

=45

=50

=55

=0

=40

=45

=50

=55

=0



86

Fig. 5.19 , 5.20 display the distribution of W concerned with 1 for different values of

and 2 . It is revealed from Fig.5.20 that the initial influence of second porous layered is

virtually minimal for both the roughness structures.


for the combination of 1 and


for the combination of 1 and 2

1.16

1.39

1.62

1.86

2.09

2.32

0.000 0.005 0.010 0.015 0.020

LO

AD

1

=40

=45

=50

=55

=0

=40

=45

=50

=55

=0

0.81

0.99

1.17

1.34

1.52

1.70

0.000 0.005 0.010 0.015 0.020

LO

AD

1

=0

=0.00019

=0.00190

=0.019

=0.19

=0

=0.0019

=0.0190

=0.19


87



A closed look at Tables 5.1 to 5.3 suggests that the first form 1G of the roughness pattern is

found to be more favorable for the adoption in the bearing system. In addition, even 2G can

be taken into consideration when the porosity is at the reduced level and magnetic strength

is in force. Although there is at least 1.8 % decrease in load bearing capacity with regards to

1G

TABLE 5.1 Change in W

with regards to different values of

W of

1G W of

2G % increase

in W

1

2

45

0.10

0.10

1

0.001

0.019

=

= −

= −

=

=

=

0.1 = 2.15390788

1.98220325

8.5

0.2 = 1.99474005

1.82296845

9.3

0.3 = 1.79837823

1.63464754

9.8

0.4 = 1.61075813

1.462575132

10.2

0.5 = 1.45201373

1.322623759

9.6

1.63

6.30

10.98

15.65

20.33

25.00

5 10 15 20 25 30

LO

AD

=0.1

=0.5

=1

=1.5

=

=0.1

=0.5

=1

=1.5

=



88

TABLE 5.2. Change in W


W of

1G W of

2G %

increase

in W

1

2

45

0.05

0.10

1

0.001

0.019

=

=

= −

=

=

=

0.20 = − 1.62101704

1.40572313

15.7

0.10 = − 1.45201373

1.32262376

9.8

0 = 1.32083663

1.24519182

6.4

0.10 = 1.21850784

1.175702334

3.4

0.20 = 1.13831751

1.115138546

1.8

TABLE 5.3 Change in W


5.4 Conclusion

The combined influence of surface roughness and ferrofluid lubrication of doubled layered

porous rough conical plate is analyzed with two different forms of transverse roughness

patterns. from the graphical results, our study discovered the following conclusions.

W

of 1G W

of 2G

% increase

in W

1

2

45

0.05

0.10

1

0.001

0.019

=

=

= −

=

=

=

0.20 = − 1.53184786

1.37329613

11

0.10 = − 1.45201373

1.32262376

9.8

0 = 1.38760648

1.27999338

8.6

0.10 = 1.33454851

1.24363128

7.2

0.20 = 1.29008292 1.212249626 6.6

5.4 Conclusion

89

The improved load due to magnetization gets sustained due to double layered.

Double layered porous conical bearing with the roughness pattern 1G is better than that of

the bearing with a roughness pattern 2G .

It has been found that the load bearing capacity remains maximum for lying between

30

to

10 approximately. (Fig 5.21).

The porosity of outer layered is favourable to develop the lubrication performance of the

double layered conical bearing. Therefore, when design the double layered porous bearing

the surface porosity should be reduced as far as possible and the roughness needs to be

treated carefully from bearing design point of view. If developed properly this investigation

may provide a good opportunity for the industry.

90

CHAPTER 6

6. Ferrofluid Lubrication of Double Layered

Rough Circular Plates with Slip Velocity

6.1 Introduction

Ferrofluid is widely used in the technical application, dynamic sealing, damping, and heat

dissipation. Ferrofluid is also used in biomedical application like hyperthermia, constant

improvement for MRI. Prakash & VIJ (1973a) analyzed the influence of squeeze films

between porous plates of different shapes .Srinivasan (1977a) examined the squeeze film of

doubled layered porous plates having different geometries like conical, circular, annular,

elliptic and rectangular. Closed form solutions are found for pressure and load capacity also

the comparison was made between traditional and double layered porous plates. The results

showed that load capacity increases due to doubled layered porous plates. Verma (1983)

presented an investigation for a double layered porous journal bearing employing short

bearing approximations. The performance characteristics were found to be improved due to

the low permeability of the inner porous layer. Rao et al. (2013) analyzed the influence of

double layered porous journal bearing lubricated couple stress fluid and Newtonian fluids.

A double layered porous lubricant film configuration with a low permeability porous layer

on top of a high permeability bearing adherent porous layer improved the bearing

performance.

Patel et al. (2011) considered the impact of rough porous circular plates with

magnetic fluid. They have considered the variable thickness for the porous matrix. It was

shown that by taking suitable thickness ratio and magnetic strength the contrary effect of

roughness could be reduced when negatively skewed roughness is in place. Sparrow et

al.(1972)studied influence of slip velocity on porous walled squeeze films. Using the slip

91

boundary condition (Patel & Deheri, 2016c) studied the combined effect of roughness and

slip velocity on the Jenkins model-based ferrofluid lubrication of a curved rough annular

squeeze film. When the slip was at a minimum, Jenkin's model-based ferrofluid lubrication

offered a method for reducing the contrary influence of roughness considering suitable

values of curvature parameter. Cusano (1972) made an analytic study of doubled-layered

porous bearing with infinite width. Results were analyzed for relating the eccentricity ratio

and coefficient friction as a function of load. Patel and Deheri (2014a) analyzed the influence

of ferrofluid for rough porous slider bearing with combined porous facings of Kozeny

Carman and Irmay. The Kozeny Carman model was found to perform better. Vadher et al.

(2008) analyzed the performance of hydromagnetic squeeze film lubrication for two

conducting porous circular plates with a rough surface. It was revealed that

hydromagnetization compensated the contrary outcome of transverse surface roughness to a

large extent with the choice of suitable plate conductivities.

Patel and Deheri (2014b) deliberated the joint influence of surface roughness and slip

velocity for curved circular plates with the Jenkins model based ferrofluid lubrication. The

Jenkins model modified the performance as related to the Neuringer – Roseinweig model

but this model provided little aid to negatively skewed roughness to augment bearing

performance. Ahamd and Singh (2007) studied the influence of slip velocity for porous

pivoted slider bearing with ferrofluid. .

6.2 Analysis

FIGURE 6.1 Configuration of the doubled layered circular plates

Figure 6.1 represents the geometry of doubled layered rough porous circular plates. The

lubricant film is filled with ferrofluid based incompressible fluid. Both the porous layered

Ferrofluid Lubrication of Double Layered Rough Circular Plates with Slip Velocity

92

are supposed to be homogeneous and isotropic. In addition to the theory of hydrodynamic

lubrication, the associated Reynold’s equation leading the film pressure is accomplished

(Srinivasan, 1977a).

31 1 2 2

1 12

12 12

d dp hr

r dr dr H H H

•

=

+ +

(6.1)

In the sight of (Neuringer Rosenweig, 1964) (6.1) transformed to

( )20 3

1 1 2 2

1 120.5

12 12

d d hr p

r dr dr H H H

•

− =

+ +

H (6.2)

Here bearing surface is considered to be transversely rough. In the context of (Christensen

& Tonder, 19969a, 1969b, 1970a) and (Beavers & Joseph ,1967) (6.2 ) is turning out to be

( )( )

20

1 120.5

d d hr p

r dr dr g h

•

− =

H (6.3)

Where

( )

1 1 2

3 3 33 2 2 2 2 3

1 1 2 2

4 4 4 44 3 2 4

2 2 2 2

12 12

sh sh sh shg h h h h h

sh sh sh sh

H H

+ + + + = + + + + + +

+ + + +

+ +

wherein

1 2

ks =

+ is the slip parameter, k is a slip coefficient

And the magnitude of the magnetic field is

( )2 Aa a r= −H , ar 0 (6.4)

Where in A is an appropriate constant reliant on the material to yield a field of preferred

magnetic strength.

The related boundary conditions are

( )0

0 and 0r

dpp a

dr =

= =

(6.5)

6.2 Analysis

93

Now integrating (6.3) and using the conditions (6.5) one gets the film pressure distribution

as

( )( )2 2 2

0

30.5

hp r a

g h

•

= − +

H (6.6)

Now load bearing capacity W is calculated from

0

2r

W rp dr=

which leads to

( )

4403

2 6

Aah aW

g h

•

−= +

(6.7)

Now, dimensionless load capacity W is found as

2 4

3h WW

h a

•= −

(6.8)

3

2 6W

G

= + (6.9)

where

30 h A

h

•

−=

, 3

2223

1113

,,,,,h

H

h

H

hhhshs

======

and

1 1 2

3 3 32 2

2 31 2

4 4 4 44 3 2

2 2 2 2

4 12 12

s s s sG

s s s s

+ + + += + + + +

+ + + +

+ + + + +


94

6.3 Results and Discussions

As the expression of W from (6.9) is linear with respect to the , it can be easily seen that

the W will be increased with increasing values of . Equation (6.9) suggests that the W

increases by 6

as associated with the traditional lubricant based bearing structure.

From Figs 6.2 - 6.5 it is evidently perceived that the W rises sharply with a rise in

. This is owing to the fact that increases lubricant’s viscosity. However, the initial

effect of both the porous layer remains insignificant.





*

0.28

0.30

0.32

0.33

0.35

0.37

0.00 0.06 0.13 0.19 0.25

LO

AD

=0 =0.05 =0.1 =0.15 =0.0

0.19

0.23

0.28

0.32

0.37

0.41

0.00 0.06 0.13 0.19 0.25

LO

AD

=−0.0 =−0.10 =0 =0.10 =0.0


95

FIGURE 6.4. Profile of W




The fact that bearing suffers heavily because of slip velocity can be seen from Figs (6.6 –

6.9). Also, from Fig. (6.7) and (6.8) it is noted that W increases as variance (-ve) increase,

while skewness follows the path of variance in this matter. Further, from Fig. (6.9) it is found

that the initial effect of 2 is almost nominal.

0.19

0.24

0.28

0.33

0.37

0.42

0.00 0.06 0.13 0.19 0.25

LO

AD

1=0 1=0.0001 1=0.001

1=0.01 1=0.1

0.13

0.19

0.25

0.30

0.36

0.42

0.00 0.06 0.13 0.19 0.25

LO

AD

=0 =0.00019 =0.0019

=0.019 =0.19


96


for the combination of *

1

s and


for the combination of 1

s and



1

s and

0.29

0.32

0.36

0.39

0.43

0.46

0.10 0.40 0.70 1.00 1.30

LO

AD

1/s*

=0 =0.05 =0.1 =0.15 =0.0

0.20

0.26

0.32

0.39

0.45

0.51

0.10 0.40 0.70 1.00 1.30

LO

AD

1/s*

=−0.0 =−0.10 =0

=0.10 =0.0

0.30

0.36

0.43

0.49

0.56

0.62

0.10 0.40 0.70 1.00 1.30

LO

AD

1/s*

=−0.0 =−0.10 =0 =0.10 =0.0


97



1

s and 2

The values of W significantly fall because of growing the values of which can be had

from Fig. (6.10) and (6.11). Also, the initial effect of the second porous layer is negligible

(Fig. (6.11)). Figures (6.12) and (6.13) indicate that the combined positive outcome of

negatively skewed roughness and variance (-ve) may be used for developing a bearing

system with enhanced performance. Here also the initial effect of the second porous layer

remains negligible. The initial influence of 1 on W with regards to is negligible (Fig.

(6.14)).



0.15

0.23

0.31

0.40

0.48

0.56

0.10 0.40 0.70 1.00 1.30

LO

AD

1/s*

=0 =0.00019 =0.0019

=0.019 =0.19

0.19

0.23

0.27

0.31

0.35

0.39

0.00 0.05 0.10 0.15 0.20

LO

AD

*

=−0. =−0.1 =0 =0.1 = 0.


98







0.15

0.20

0.25

0.31

0.36

0.41

0.00 0.05 0.10 0.15 0.20

*

=0 =0.00019 =0.0019

=0.019 =0.19

0.20

0.24

0.29

0.33

0.38

0.42

-0.20 -0.10 0.00 0.10 0.20

LO

AD

*

=−0. =−0.1 =0 =0.1 = 0.

0.13

0.20

0.27

0.33

0.40

0.47

-0.20 -0.10 0.00 0.10 0.20

LO

AD

*

=0 =0.00019 =0.0019

=0.019 =0.19

6.4 Conclusion

99



Some of the Figures send the message that with an appropriate magnetic strength the

combined optimistic influence of negatively skewed roughness and variance (-ve) may be

counted for neutralizing the contrary influence of slip velocity and porosity.

6.4 Conclusion

This study makes it clear that for enhanced performance the slip velocity is needed to keep

at reduce level.

The roughness features must be focused prudently while designing the bearing system.

The magnetization presents a limited scope in easing the contrary impact of roughness,

porosity combines even if the slip is at a diminished level. In spite of the contrary influence

of many parameters, the bearing system sustains a good amount of load in the absence of

flow, which does not occur in the non-magnetic case.

0.14

0.20

0.26

0.32

0.38

0.44

-0.20 -0.10 0.00 0.10 0.20

LO

AD

*

=0 =0.00019 =0.0019=0.019 =0.19

100

CHAPTER 7

7. Ferrofluid Based Longitudinally Rough

Porous Circular Stepped Plates in the

Existence of Couple Stress

7.1 Introduction

This chapter directs to scrutinize the impact of ferrofluid in the presence of couple stress for

longitudinally rough porous circular stepped plates. The influence of longitudinal surface

roughness is developed using the stochastic model of Christensen and Tonder for nonzero

mean, variance and skewness. Neuringer-Roseinweig model is adopted for the influence of

ferrofluid. The couple stress effect is characterized by Stokes micro continuum theory. The

closed form solutions for load bearing capacity and film pressure are obtained as a function

of different parameters and plotted graphically. It is perceived that load capacity gets

improved owing to the combined influence of magnetization and couple stress when the

proper choice of roughness parameters (negatively skewed, standard deviation) are in place.

A ferrofluid is a liquid that contains a colloidal suspension of ferromagnetic particles

and that becomes strongly magnetized in the existence of an external magnetic field. Use of

ferrofluid as a lubricant is found in bearings, loudspeakers, dampers, sensor as well as in

biomedical instruments. Several investigators have also attempted to discover its use as a

lubricant in squeeze film bearing structures. Verma (1986) studied the influence of ferrofluid

based squeeze film lubrication on two approaching surfaces and concluded that magnetic

fluid-based squeeze film is better than the vicious squeeze film. Bhat and Deheri (1993)

studied squeeze film characteristic in the curved circular disk by considering the magnetic

effect and found that bearing’s load capacity enhances due to magnetization parameter. Shah

101

(2003) analyzed the impact of ferrofluid lubrication in step bearing. Dinesh Kumar

et. al.(1992) considered the influence of ferrofluid on spherical and conical bearings using

perturbation analysis. Shah and Bhat (2005)studied the impact of magnetic fluid on curved

annular plates. They considered the revolution of magnetic particles and their magnetic

moments. Their study revealed that load capacity and response time of bearing improved

due to Langevin’s parameter. Agrawal (1986) examined the impact of a porous inclined

slider bearing with magnetic fluid and deduced that bearing’s life span is grater compare to

viscous porous inclined slider bearing. Shah and Bhat (2003) analyzed the impact of

ferrofluid on the parallel plate squeeze film bearing.

Additives have been added in the fluid to create the flow properties and to enhance

the lubricating qualities. Couple stress and micropolar fluids are examples of such types of

fluids, which has become more important for current industrial materials. Extensive studies

have been carried out to describe the importance of couple stress fluid in different bearing

geometries. Ramanaiah and Sarkar (1978) analyzed the upshot of couple stress for the thrust

bearing. Lin (1998) studied finite journal bearing. Maiti (1973) analyzed the performance

characteristic of the composite slider bearing with micropolar fluid. Lin et al (2006) studied

the performance of couple stress fluid based wide parallel plates. Naduvinamani and

Siddangouda (2009) considered a couple stress fluid to study the impact of squeeze film

lubrication in circular stepped plates. In all these investigations the couple stress impact is

governed by (Stokes,1966) microcontinuum theory and all these investigations discovered

the importance of couple stress fluid compared to Newtonian lubricants such as improved

bearing’s load capacity, reduced coefficient of friction and growth in squeeze film time.

However, it is well known that after having some run-in wear bearing surfaces

develop some roughness so, surface roughness and its impact on bearing performance have

been deliberated by various investigators. Some mathematical models have been anticipated

in the derivation of Reynolds type equations which accounts for surface roughness effect.

Among all these models stochastic approach given by (Christensen & Tonder, 1969a, 1969b,

1970a) is used very widely. Many investigators have studied the combined influence of

surface roughness and ferrofluid for distinct porous bearing configuration (Shimpi & Deheri,

2012; Patel et al., 2011; Shukla & Deheri, 2013; Vashi et al., 2018). All these scrutinizes

discovered that the load capacity of the bearing improves owing to ferrofluid and negatively

skewed roughness. Many authors have analyzed the combined influence of surface

roughness and couple stress on different bearing geometry (Naduvinamani & Siddangouda,

2007; Siddangouda, 2015a; Naduvinamani & Biradar, 2006; Bujurke et al., 2008;

Ferrofluid Based Longitudinally Rough Porous Circular Stepped Plates in the Existence of

Couple Stress

102

Naduvinamani et al., 2012 ) and their investigations confirmed that load capacity and

squeeze film time increases owing to non-Newtonian behaviour of fluid compared to

Newtonian case.

7.2 Analysis

Fig. 7.1 displays the bearing geometry. The lower plate has porous facing which remains

fixed while the upper plate is moving with squeeze velocity V . The fluid region is filled

with incompressible ferrofluid with a couple stress effect.

FIGURE 7.1 The physical geometry of longitudinally rough circular stepped plates

Through the traditional traditions of hydrodynamic lubrication, the governing Reynolds type

equation for the film pressure turns out to be (Naduvinamani & Siddangouda, 2009)

( )6

,

i

i i

dp V r

d r S H l

−=

(7.1)

7.2 Analysis

103

( ) 3 2 3, 12 12 242

ii i i i

HS h l H H l H l tanh

l

= + − +

(7.2)

for smooth bearing, to derive the longitudinal surface roughness, we adopted the stochastic

approach given by (Christensen & Tonder, 1969a, 1969b, 1970a) in equation (7.1).

( )6

,

i

i i

dp Vr

dr g h l

−=

(7.3)

Where,

( )( ) ( )

( )1

2 303 1

1

1 1, 12 12 24

2

i

i i

i i

E Hg h l H l l tanh

lE H E H

−

− −

= + − +

(7.4)

Where,

( )( ) ( )

( )2 30

1

, , ,1 1, , , , 12 12 24

, , , , , , 2

i ii i

i i i i

n hg h l H l l tanh

m h n h l

= + − +

(7.5)

( ) ( ) ( )( )3 1 2 2 2 3 2 3, , , 1 3 6 10 3i i i i i im h h h h h− − − −= − + + − + + (7.6)

( ) ( ) ( )( )1 1 2 2 2 3 2 3, , , 1 3i i i i i in h h h h h− − − −= − + + − + + (7.7)

Neuringer-Rosenswein (1964) intended a simple model to define the stable flow of

ferrofluids in the existence of gradually varying magnetic fields. Therefore, in the context of

(Neuringer & Rosensweig 1964) model, additional pressure term 2

0

1

2 H is present in the

equation of motion when ferrofluid is used as a lubricant. Hence (7.3) transfer to

( )

20

120.5

, , , ,i

i i

d Vrp

dr g h l

− − =

H (7.8)

wherein (7.14) 2H denotes the magnetic field’s magnitude and is defined as

( )( )2 A R r r KR= − −H (7.9)


Couple Stress

104

wherein A is an appropriate constant reliant on the material to yield a field of expected

magnetic field strength. Where,

1ih h= minimum film thickness in the film region 0 r KR and

2ih h= maximum film thickness in the film region KR r R

The related fluid film pressure boundary conditions are

1 2p p= at r KR= and (7.10)

2 0p = at r R= (7.11)

By taking the integration of (7.8) with respect to r and making use of (7.10) and (7.11)

yields pressure in both the film regions

( )

( )( )

( )2 2 2 2 2 2 21 0

1 1 2 2

3 30.5

, , , , , , , ,

V Vp K R r R R K

g h l g h l

= − + − + H (7.12)

( )( )2 2 2

2 0

2 2

30.5

, , , ,

Vp R r

g h l

= − + H (7.13)

Where,

( )( ) ( )

( )1 12 31 1 0

1 1 1 1

1

, , ,1 1, , , , 12 12 24

, , , , , , 2

n hg h l H l l tanh

m h n h l

= + − +

(7.14)

( )( ) ( )

( )2 22 32 2 0

2 2 2 2

1

, , ,1 1, , , , 12 12 24

, , , , , , 2

n hg h l H l l tanh

m h n h l

= + − +

(7.15)

Using (7.12) and (7.13) the load bearing capacity W is obtained as

1 20

2 2KR R

KR

W p rdr p rdr = + (7.16)

7.2 Analysis

105

which yields

( )( )

( )( )

44 14 40.5 30 1 26 2 , , , ,, , , , 2 21 1

KR A VR KW K

g h lg h l

−

= − + +

(7.17)

By making the use of following dimensionless variables in (7.17)

3* * * * * *0 2 01

3 32 2 2 2 2 2

2- , , , , , ,

Ah Hh lH l

V h h h h h h= = = = = = =

Expression (7.17) transfer to dimensionless load capacity as follow.

( ) ( )43 4

2

41 2

12 12

183

KKwh KW

G GVR

−− = = + +

(7.18)

Where,

( )

( )

232

1

1 1 1 1

3

1

1 312

( , , , , ) ( , , , ) , , ,

13

, , ,

h lG

g h l M H N H

l tanhN H l

= = + − +

( )

32

2

2 2 , , , ,

hG

g h l = =

( ) ( )

23

2 2 2

1 3 112 3

( , , ) , , , ,

ll tanh

M N N l

+ − +

Where,

( ) ( ) ( )( )3 1 2 2 2 3 2 31 , , , 1 3 6 10 3M H H H H H − − − − = − + + − + +

( ) ( ) ( )( )* * * * 1 * *-1 *-2 *2 *2 *-3 *2 *3 *1 , , , 1- - 3N H H H H H− = + + + +

( ) ( ) ( )( )2 2 2 32 , , 1 3 6 10 3M = − + + − + +

( )2 2 2 3

2 , , 1 3N = − + + − + +


Couple Stress

106


This study examined the performance characteristic of longitudinally rough porous circular

stepped plates with ferrofluid in the presence of couple stress effect. Equation (7.18) suggest

that load capacity enhances by ( )2 1

18

K − times as compared to conventional lubricant

based bearing system. Also, the comparison is made between the ferrofluid based bearing

system and couple stress fluid-based bearing system. The variation of W with regards to

for distinct values of , , , ,K l is displayed in Fig. 7.2 - 7.7. One can notice from

all these Figs. that the outcome of is to improve the load bearing capacity. Fig. 7.7 shows

that Wattain its maximum value when 0.5l = and 0.25 = .

FIGURE 7.2 Profile of W for the combination of and K

FIGURE 7.3 Profile of W for the combination of and *

0.89

1.05

1.21

1.36

1.52

1.68

0.00 0.06 0.13 0.19 0.25

LO

AD

K=0.45 K=0.55 K=0.65 K=0.75 K=0.85

1.43

1.51

1.58

1.66

1.73

1.81

0.00 0.06 0.13 0.19 0.25

LO

AD

= 0 = 0.05 = 0.10 = 0.15 = 0.


107

FIGURE 7.4 Profile of W for the combination of and *



1.19

1.27

1.35

1.42

1.50

1.58

0.00 0.06 0.13 0.19 0.25

LO

AD

=−0.05 =−0.05 =0

=0.05 = 0.05

0.55

0.80

1.06

1.31

1.57

1.82

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=−0.05 =−0.05 =0 =0.05 = 0.05

0.47

0.67

0.88

1.08

1.29

1.49

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1


Couple Stress

108


Figures 7.8-7.12 represent the influence of K on load capacity for various values of

, , , and l. From all these Figures one can perceive that W

falls down as

increasing the values of K . Also, Figure 7.12 suggests that the reduction in W is nominal

in case of couple stress parameter.


1.13

1.34

1.54

1.75

1.95

2.16

0.00 0.05 0.10 0.15 0.20 0.25

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

1.15

1.47

1.78

2.10

2.41

2.73

0.45 0.55 0.65 0.75 0.85

LO

AD

K

= 0 = 0.05 = 0.10 = 0.15 = 0.


109




0.73

0.95

1.16

1.38

1.59

1.81

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0.05 =−0.05 =0 =0.05 = 0.05

0.37

0.72

1.07

1.41

1.76

2.11

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=−0.05 =−0.05 =0 =0.05 = 0.05

0.32

0.60

0.88

1.16

1.44

1.72

0.45 0.55 0.65 0.75 0.85

LO

AD

K

=0 =0.0001 =0.001 =0.01 =0.1


Couple Stress

110

FIGURE 7.12 Profile of W for the combination of K and l

Distribution of W with regards to H for distinct values of , , ,

and lis

displayed in Fig. 7.13 - 7.17. It is perceived that W falls down as increasing the value of

H . It is clearly noticed from these Figs. that decrease in Wis nominal when H surpasses

the value 2.26.

FIGURE 7.13 Profile of W for combination of H and

0.71

1.07

1.43

1.78

2.14

2.50

0.45 0.55 0.65 0.75 0.85

LO

AD

K

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

1.41

1.51

1.61

1.70

1.80

1.90

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

= 0 = 0.05 = 0.10 = 0.15 = 0.


111




1.17

1.27

1.37

1.46

1.56

1.66

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0.05 =−0.05 =0 =0.05 = 0.05

0.54

0.82

1.09

1.37

1.64

1.92

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=−0.05 =−0.05 =0 =0.05 = 0.05

0.46

0.68

0.91

1.13

1.36

1.58

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

=0 =0.0001 =0.001 =0.01 =0.1


Couple Stress

112

FIGURE 7.17 Profile of W for the combination of H and l

Figures 7.18 -7.21 characterize the positive impact of on load capacity for various values

of , l . All these Fig. enlighten that there is a sharp growth in load cpapcity as the

values of increases. Figure 7.18 suggests that maximum load is registered when 0.5l =

and 0.2 = .


1.12

1.35

1.58

1.82

2.05

2.28

1.30 1.62 1.94 2.26 2.58 2.90

LO

AD

H*

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5

1.18

1.34

1.49

1.65

1.80

1.96

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0.05 =−0.05 =0

=0.05 = 0.05


113




0.55

0.88

1.21

1.54

1.87

2.20

0.00 0.05 0.10 0.15 0.20

LO

AD

=−0.05 =−0.05 =0 =0.05 = 0.5

0.47

0.75

1.02

1.30

1.57

1.85

0.00 0.05 0.10 0.15 0.20

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

1.12

1.47

1.82

2.17

2.52

2.87

0.00 0.05 0.10 0.15 0.20

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


Couple Stress

114

The positive influence of and

on load capacity with regards to and l can hold

from Fig. 7.22-7.25. From these Figures one can observe that a negatively increase in the

values of and

rises the dimensionless load capacity of bearing. So, the optimistic

impact of and

maybe suitably considered while designing the bearing structure.

Figures 7.6,7.11, 7.16,7.20 and 7.22 give a message that the initial impact of porous facing

is insignificant up to 0.001= .



0.32

0.63

0.94

1.26

1.57

1.88

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

0.52

1.03

1.55

2.06

2.58

3.09

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


115

FIGURE 7. 24 Profile of W for the combination of and

FIGURE 7. 25 Profile of W for the combination of and l


R for distinct values of K and l

K l

W for the present study with

0.15, 0.15, 0.025, 0.025,

0.001, 2.10H

= = = − = − = =

W

with traditional

lubrication

( )0, 2.10H = =

Relative

percentage

growth in

W

( )W

R

0.45

0.1 1.3196295652

0.99006180 33.29

0.3 1.6773566087

1.18745642 41.26

0.5 2.4927471676

1.57332509 58.44

0.65

0.1 1.1502444427

0.86369031 33.18

0.3 1.4575267267

1.03353911 41.02

0.5 2.1576304076

1.36552978 58.01

0.85

0.1 0.7220956700

0.54799135 31.77

0.3 0.9033577730

0.64902567 39.19

0.5 1.3154538713

0.84641930 55.41

0.44

0.67

0.91

1.14

1.38

1.61

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

=0 =0.0001 =0.001 =0.01 =0.1

0.96

1.24

1.51

1.79

2.06

2.34

-0.05 -0.03 -0.01 0.01 0.03 0.05

LO

AD

l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5


Couple Stress

116


R for distinct values of H and l

H

l

W for the present study with

0.15, 0.15, 0.025, 0.025,

0.001, 0.65K

= = = − = − = =

W with traditional

lubrication

( )0, 0.65K = =

Relative

percentage

growth in W

( )W

R

1.3

0.1 1.23184347 0.92689311 32.90

0.3 1.55161185 1.10583213 40.31

0.5 2.27664670 1.45544961 56.42

2.1

0.1 1.15524444 0.86369031 33.76

0.3 1.46252673 1.03353911 41.51

0.5 2.16263041 1.36552978 58.37

2.9

0.1 1.14218948 0.85163399 34.12

0.3 1.44862282 1.02073256 41.92

0.5 2.14709608 1.35128724 58.89

Tables 7.1 - 7.2 bring to light that bearing performance improves due to the combined

influence of ferrofluid and couple stress compared to couple stress fluid-based bearing

structure. It is perceived from Table 7.1 that there is a growth of about 58% in Wwhen

0.65K = and 0.5l = . Also, it is seen from Table 7.2 that the relative increase in W is

around 59% when 2.9H = and 0.5l = .

7.4 Conclusion

The combined impact of surface roughness and ferrofluid fluid for porous circular stepped

plates in the existence of couple stress is analyzed. Based on above results and discussion

following conclusions are made.

The graphical and tabular results show that ferrofluid fluid as a lubricant offers increased

load bearing capacity compared to the non-magnetic case.This increase in load is 58%

greater compared to non magnetic case.

Further, this development in load capacity is enhanced in the presence of standard deviation

and negatively skewed roughness with a couple stress parameter.

The reduction in load is owing to porous facing and positively skewed roughness can be

compensated by employing the ferrofluid as a lubricant with the suitable choice of a couple

stress parameter through which the life span of circular step bearing can be boosted.

117

CHAPTER 8

8. General Conclusions and Future Scope of The

Work

8.1 General Conclusions

The present thesis examine the combined influence of surface roughness and ferrofluid

lubrication on bearing’s load capacity. Modified Reynolds equation leading to pressure

distribution is stochastically averaged using polynomial probability distribution function

with regards to roughness parameter correspond to non zero mean, standard deviation and

skewness. The standard deviation can take only positive values while mean and skewness

assumes both the values positive and negative. The influence of ferrofluid and surface

roughness are analyzed for various geometries of the bearing. The load bearing capacity is

boosted by ferrofluid lubrication.

From all investigations presented in the thesis concluded that with respect to

transverse roughness all the roughness parameters affect the bearing system adversely, while

in the case of longitudinal roughness we can observe from the present analysis that mean

and skewness tend to drop the load capacity and a noticeable fact is that standard deviation

growth the bearing’s load capacity.

Roughness and porosity affect the bearing system adversely. The investigation for

parallel plates bearing suggests that from the design point of view position of step and

roughness features play an important role. The adverse influence of roughness and porosity

can be compensated with proper selection of magnetization parameter and couple stress

parameter when variance (-ve) is in place. Analysis of double layered porous bearing

suggests that the porosity of the outer layer influences more as compared to the inner layer

even in the presence of mild magnetic strength. Obtained results are compared with the

118

nonmagnetic case and found that load bearing capacity improves in the case of

magnetic case compared to the non-magnetic case.

8.2 Future Scope of the Work

The current study opens up a new extent of research and improvement in several directions

as follows:

▪ In the present study, Neuringer -Rosenweig model for ferrofluid lubrication is used. The

influence of surface roughness may be studied with Shliomis and Jenkin's model for

ferrofluid lubrication.

▪ The influence of bearing deformation can be studied for various types of bearing.

Investigation can be made by using porous models of Kozeny-Carman’s and Irmay’s.

▪ The effect of transverse and longitudinal roughness can be considered for multi-stepped

bearings. Additionally, the results obtained in this thesis may be extended to more

complex geometries associated with practical applications.

▪ The investigation may be carried out by considering the Rabinowitsch fluid model for

different bearing geometries.

119

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List of Publications

List of Publications

1. Ferrofluid based squeeze film lubrication between rough stepped plates with couple

stress effect. Journal of Applied Fluid Mechanics, 11(3), 597-612, 2018. doi.org/

10.29252/jafm.11.03.27854

2. Combined influence of ferrofluid and longitudinal roughness on porous parallel

stepped plates with couple stress. International Journal of Research in Advent

Technology, , 7(2), 584-592, 2019. doi.org/10.32622/ijrat.72201913.

3. Combined effect of slip velocity and surface roughness on the ferrofluid based squeeze

film lubrication in double layered porous circular plates. Global Journal of Pure and

Applied Mathematics, 13(9), 5367-5380 ,2017.

4. Load bearing capacity for a ferrofluid squeeze film in double layered porous rough

conical plates. In: K.N.Das et al. (eds)Proceeding of 8th Interntional conference on

soft computing for problem solving-SocPros 2018,VIT-Vellore,Tamil

Nadu,India,17-19 December 2018. Advances in Intelligent Systems and Computing

series of Springer, Singapore,1048, 9-25, 2020. doi.org/10.1007/978-981-15-0035-

0_2.

.

https://www.doi.org/%2010.29252/jafm.11.03.27854

https://www.doi.org/%2010.29252/jafm.11.03.27854

http://www.ijrat.org/downloads/Vol-7/feb-2019/Paper%20ID-72201913

https://link.springer.com/chapter/10.1007%2F978-981-15-0035-0_2

https://link.springer.com/chapter/10.1007%2F978-981-15-0035-0_2

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