Numerical Modeling Based Analysis of - Amazon S3 · Numerical Modeling Based Analysis of Hydrodynamic Sliding Contact Bearings A Thesis submitted to Gujarat Technological University

Numerical Modeling Based Analysis ofHydrodynamic Sliding Contact Bearings

A Thesis submitted to Gujarat Technological University

for the award of

Doctor of Philosophyin

Science-Mathsby

Mehul P. PatelEnrollment No.: 139997673010

under supervision of

Dr. Himanshu C. Patel

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

March 2019

[email protected]

Faculty Web Site URL Here (include http://)

Numerical Modeling Based Analysis ofHydrodynamic Sliding Contact Bearings

A Thesis submitted to Gujarat Technological University

for the award of

Doctor of Philosophyin

Science-Mathsby

Mehul P. PatelEnrollment No.: 139997673010

under supervision of

Dr. Himanshu C. Patel

GUJARAT TECHNOLOGICAL UNIVERSITY

AHMEDABAD

March 2019

c©Mehulkumar Prabhuram Patel

ii

DeclarationI declare that the thesis entitled "Numerical Modeling Based Analysis of

Hydrodynamic Sliding Contact Bearings" submitted by me for the degree of Doctor

of Philosophy is the record of research work carried out by me during the period from

February 2014 to December 2017 under the supervision of Prof. Dr. H. C. Patel,

Professor and Head, Department of Mathematics, L.D. College of Engineering,

Ahmedabad, Gujarat and this has not formed the basis for the award of any degree,

diploma, associateship, fellowship, titles in this or any other University or other

institution of higher learning.

I further declare that the material obtained from other sources has been duly

acknowledged in the thesis. I shall be solely responsible for any plagiarism or other

irregularities, if noticed in the thesis.

Signature of the Research Scholar: Date: 02/03/2019

Name of Research Scholar: Mehul P. PatelPlace: Patan.

iii

CertificateI certify that the work incorporated in the thesis "Numerical Modeling Based

Analysis of Hydrodynamic Sliding Contact Bearings" submitted by Shri

Mehulkumar Prabhuram Patel was carried out by the candidate under my 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 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: 02/03/2019

Name of Supervisor: Dr. Himanshu C. PatelPlace: Ahmedabad.

iv

Course-work Completion CertificateThis is to certify that Mr. Mehulkumar Prabhuram Patel, enrolment no.

139997673010 is a PhD scholar enrolled for PhD program in the branch

Science-Maths of Gujarat Technological University, Ahmedabad.

(Please tick the relevant option(s))

He has been exempted from the course-work (successfully completed during

M.Phil Course)

He has been exempted from Research Methodology Course only (successfully

completed during M.Phil Course)

He has successfully completed the PhD course work for the partial requirement

for the award of PhD Degree. His performance in the course work is as follows-

Grade Obtained in Research Methodology Grade Obtained in Self Study Course (Core Subject)

(PH001) (PH002)

CC AB

Supervisor’s Sign:

Name of Supervisor: Dr. Himanshu C. Patel

v

Originality Report CertificateIt is certified that PhD Thesis titled "Numerical Modeling Based Analysis of

Hydrodynamic Sliding Contact Bearings" by Shri Mehulkumar Prabhuram Patel

has been examined by us. We undertake the following:

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

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

table, paragraph or section has been copied verbatim from previous work unless it is

placed under quotation marks and duly referenced.

b. The work presented is original and own work of the author (i.e. there is no

plagiarism). No ideas, processes, results or words of others have been presented as

Author own work.

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d. There is no falsification by manipulating research materials, equipment or

processes, or changing or omitting data or results such that the research is not

accurately represented in the research record.

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 ≤ 25%).

Signature of the Research Scholar: Date: 02/03/2019

Name of Research Scholar: Mehul P PatelPlace: Patan.

Signature of Supervisor: Date: 02/03/2019

Name of Supervisor: Dr. Himanshu C. PatelPlace: Ahmedabad.

vi

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Numerical Modeling Based Analysis of Hydrodynamic Sliding

Contact Bearings

ORIGINALITY REPORT

PRIMARY SOURCES

"Proceedings of International Conference on

Advances in Tribology and Engineering

Systems", Springer Nature America, Inc, 2014Publicat ion

www.tribology.fink.rsInternet Source

Submitted to Universiti Teknologi MalaysiaStudent Paper

"Proceedings of the 1st International

Conference on Numerical Modelling in

Engineering", Springer Nature America, Inc,

2019Publicat ion

Submitted to School of Business and

Management ITBStudent Paper

Submitted to Monash University Sunway

Campus Malaysia Sdn BhdStudent Paper

vii

PhD THESIS Non-Exclusive License to

GUJARAT TECHNOLOGICAL

UNIVERSITYIn consideration of being a PhD Research Scholar at GTU and in the interests of the

facilitation of research at GTU and elsewhere, I, Mehulkumar Prabhuram Patel, having

enrolment no. 139997673010, hereby grant a non-exclusive, royalty free and perpetual

license to GTU on the following terms:

a). GTU is permitted to archive, reproduce and distribute my thesis, in whole or in

part, and/or my abstract, in whole or in part (referred to collectively as the "Work")

anywhere in the world, for non-commercial purposes, in all forms of media;

b). GTU is permitted to authorize, sub-lease, sub-contract or procure any of the acts

mentioned in paragraph (a);

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the authority of their "Thesis Non-Exclusive License";

d). The Universal Copyright Notice ( c©) shall appear on all copies made under the

authority of this license;

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Archives. Any abstract submitted with the thesis will be considered to form part of

the thesis.

f). I represent that my thesis is my original work, does not infringe any rights of others,

including privacy rights, and that I have the right to make the grant conferred by this

non-exclusive license.

g). If third party copyrighted material was included in my thesis for which, under the

terms of the Copyright Act, written permission from the copyright owners is

required, I have obtained such permission from the copyright owners to do the acts

mentioned in paragraph (a) above for the full term of copyright protection.

viii

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.

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:

Name of Research Scholar: Mehul P. PatelDate: 02/03/2019 Place: Patan.

Signature of Supervisor:

Name of Supervisor: Dr. Himanshu C. PatelDate: 02/03/2019 Place: Ahmedabad.

Seal:

ix

Thesis Approval FormThe viva-voce of the PhD Thesis submitted by Shri Mehulkumar Prabhuram Patel

(Enrollment No. 139997673010) entitled "Numerical Modeling Based Analysis of

Hydrodynamic Sliding Contact Bearings" was conducted on 02/03/2019 at Gujarat

Technological University.

(Please tick any one of the following option)

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.

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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

x

AbstractEssentially Tribology is the study of wear and friction of inter acting surfaces in

relative motion. This study plays an important role in the bearing system in view of

manufacturing reasons as far as industries are concerned.

Tribology covers all phenomena that occur on frictional surfaces and is related closely

to various scientific fields such as mechanical engineering, physics, mathematics,

chemistry and material science. For understanding such phenomena, mathematical

model based numerical analysis may be helpful to us in the context of statistical

development. The prediction of lubricating film characteristic is the crucial factor from

application point of view.

By now, it is a well-known fact that after having some run in and wear the bearing

surfaces develops roughness. Sometimes the contamination of the lubricants and

chemical degradation of the surfaces contribute to roughness. It is well established that

the roughness has an adverse effect on the performance of the bearing system. When

the order of roughness becomes more, the life span of the bearing system gets

drastically reduced. Several methods have been proposed to reduce the negative effect

of surface roughness on the performance of bearing systems. One such method is the

replacement of conventional lubricants by a magnetic fluid lubricant. This lubricant

has many interesting properties including the one that the magnetic fluid can be

retained at a desired location by an external magnetic field. Further, the magnetic fluid

can be made to move with the help of a magnetic field gradient even in the regions

where there is no gravity.

Thus, an effort will be made to analyze the magnetization effect on the performance of

a bearing system of various shapes. Further, up to which extent the magnetic fluid can

go for minimizing the adverse effect of roughness will be the matter of investigations.

The method will be based on serving the associated statistically averaged Reynolds’

type equation to obtain the PD, which will in turn give in turn LSC resulting in the

calculation of friction.

xi

Acknowledgement

I am very much thankful to ALMIGHTY for giving me an opportunity to undertakethe research work and enabling me to its completion.First and foremost, I would like to thank my research guide Dr. Himanshu C. Patel forhis valuable guidance, scholarly inputs and consistent encouragement I receivedthroughout the research work. This feat was possible only because of the unconditionalsupport provided by him. A person with an amicable and positive disposition, He hasalways available to clarify my doubts despite of his busy schedules and I consider it asa great opportunity to do my doctoral programme under his guidance and to learn a lotfrom him.I sincerely extend my gratitude towards Dr. G. M. Deheri, Former Associate Professor,S. P. University, Vallabh Vidyanagar, Anand for his valuable guidance and usefuldiscussions at every stage of the work reported in this thesis. His constantencouragement and continuous efforts to implant interest in the subject have been ofimmense value to him.I wish to express my gratitude to Dr. Nimeshchandra S. Patel, Assistant Professor,Department of Mechanical Engineering, D. D. University, Nadiad, for his kind helpand guidance in understanding the mechanical phenomenon throughout my researchjourney.I express my thanks to my D.P.C. members Prof. D. V. Bhatt, Professor, Department ofMechanical Engineering, SVNIT, Surat and Prof. H. R. Kataria, Professor and Head,Department of Mathematics, Faculty of Science, M. S. University, Baroda, for theirextreme kindness and providing all possible facilities in their departments during myD.P.C.I would like to thank my friend Pankaj Yadav for inspiring and supporting me for thistask. Mr. Yadav is one of the most competent persons I know and having him as acolleague motiveted me to conduct the highest quality of thesis work.Finally, I am extremely thankful to my Parents, wife, child and family for providingme support during the ups and downs of this challenging period. I am grateful to thealmighty for giving me the direction and enthusiasm during the entire tenure ofresearch work.

Mehul P. Patel

xii

Contents

Abstract xi

Acknowledgement xii

List of Symbols xv

List of Figures xvii

List of Tables xix

1 Introduction 11.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Definition of the problem . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Objective and Scope of work . . . . . . . . . . . . . . . . . . . . . . . 61.4 Original contribution by the thesis . . . . . . . . . . . . . . . . . . . . 71.5 Methodology of Research and Results/Comparisons . . . . . . . . . . . 71.6 Achievements with respect to objectives . . . . . . . . . . . . . . . . . 81.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 A Prerequisite for Tribology 102.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Dry Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.2 Fluid friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.3 Lubricated Friction . . . . . . . . . . . . . . . . . . . . . . . . 122.1.4 Skin friction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Lubrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Hydrodynamics Lubrication . . . . . . . . . . . . . . . . . . . 132.3.2 Hydrostatic Lubrication . . . . . . . . . . . . . . . . . . . . . 132.3.3 Boundary Lubrication . . . . . . . . . . . . . . . . . . . . . . 132.3.4 Mixed Lubrication . . . . . . . . . . . . . . . . . . . . . . . . 142.3.5 Elasto-hydrodynamic Lubrication . . . . . . . . . . . . . . . . 14

2.4 Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.1 Hydrodynamic Bearings . . . . . . . . . . . . . . . . . . . . . 152.4.2 Hydrostatic Bearings . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Tribological Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.6 Lubricant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.7 Extension of the Classical Theory . . . . . . . . . . . . . . . . . . . . 17

2.7.1 Basic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 172.8 Slider bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

xiii

Contents xiv

2.8.1 Infinitely long slider bearing . . . . . . . . . . . . . . . . . . . 202.9 Journal bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9.1 Infinitely short journal bearing . . . . . . . . . . . . . . . . . . 212.10 Magnetic fluid lubrication . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.10.1 Neuringer-Rosensweig Model . . . . . . . . . . . . . . . . . . 232.10.2 Modified Reynolds equation of infinitely long slider bearing for

magnetic fluid . . . . . . . . . . . . . . . . . . . . . . . . . . 242.10.3 Modified Reynolds equation of infinitely short journal bearing

for magnetic fluid . . . . . . . . . . . . . . . . . . . . . . . . . 242.11 Surface Roughness Effect . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Magnetic Fluid Based an Infinitely Long Transversely Rough SliderBearing 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Magnetic Fluid Based a Short Transversely Rough Journal Bearing 404.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 A Magnetic Fluid Based a Longitudinal Rough Exponential Slider Bearing 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Several PDF Related With the Roughness Characteristics on thePerformance of Longitudinal Rough Slider Bearing 736.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 816.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

References 90

List of Publications 94

List of Symbols

List of Symbolsl Length of slider bearing (m)

U Velocity of bearing surfaces inX-Direction (ms−1)

h Nominal film thickness (m)

h0 Minimum film thickness (m)

h1 Maximum film thickness (m)

δ Random roughness amplitudes ofthe two surfaces measured from their mean level (m)

H Stochastic film thickness (m)

E() Expectancy operatorM Magnitude of the Magnetic field (Am−1)

σ Standard deviation of random surfaceroughness (m)

σ∗ Standard deviation of random surfaceroughness (Dimensionless)

α Mean of random surface roughness (m)

α∗ Mean of random surfaceroughness (Dimensionless)

ε Skewness of random surfaceroughness (m3)

ε∗ Skewness of random surfaceroughness (Dimensionless)

η Viscosity of lubricant (Kgm−1s−1)

µ∗ Magnetization parameter (Dimensionless)µ0 Magnetic susceptibility (Proportionality Constant)µ Free space permeability

(4π×10−7 KgmA−2s−2)

p Mean pressure level (Nm−2)

P Mean pressure level (Dimensionless)w Load supporting capacity (N)

W Load supporting capacity (Dimensionless)

xv

List of Symbol sand Abbreviations xvi

R j Radius of journal (m)

Rb Radius of bearing (m)

O j Centre of journalOb Centre of bearingc Radial clearence (m)

ε Eccentricity ratioc Surface roughness height (m)

c∗ Surface roughness height (Dimensionless)τ Film thickness ratio (Dimensionless)F Friction (Dimensionless)µ Coefficient of friction(Dimensionless)

AbbreviationsPD Pressure DistributionLSC Load Supporting CapacityDL Dimension LessSD Standard DeviationSB Slider BearingJB Journal BearingFT R Film Thickness RatioRPP Roughness Pattern ParameterLR Longitudinal RoughnessT R Transverse RoughnessPDF Probability Density Function

List of Figures

2.1 Slider Bearing and Journal Bearing . . . . . . . . . . . . . . . . . . . . 152.2 Hydrostatic Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Infinitely Long Slider Bearing . . . . . . . . . . . . . . . . . . . . . . 202.4 Infinitely Short Journal Bearing . . . . . . . . . . . . . . . . . . . . . 21

3.1 Infinitely Long Slider Bearing . . . . . . . . . . . . . . . . . . . . . . 293.2 DL LSC versus mean α∗ for different values of SD σ∗ . . . . . . . . . 343.3 DL LSC versus mean α∗ for different values of skewness ε∗ . . . . . . 353.4 DL LSC versus SD σ∗ for different values of skewness ε∗ . . . . . . . 363.5 DL LSC versus FTR τ for different values of skewness ε∗ . . . . . . . . 37

4.1 Short Journal Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 DL LSC versus magnetic parameter µ∗ for different values of mean α∗ 464.3 DL LSC versus magnetic parameter µ∗ for different values of skewness

ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.4 DL LSC versus magnetic parameter µ∗ for different values of SD σ∗ . . 484.5 DL LSC versus magnetic parameter µ∗ for different values of

eccentricity ratio ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.6 DL LSC versus SD σ∗for different values of mean α∗ . . . . . . . . . . 504.7 DL LSC versus SD σ∗for different values of skewness ε∗ . . . . . . . . 514.8 DL LSC versus mean α∗ for different values of skewness ε∗ . . . . . . 524.9 DL LSC versus eccentricity ratio ε for different values of mean α∗ . . . 534.10 DL LSC versus eccentricity ratio ε for different values of SD σ∗ . . . . 544.11 DL LSC versus eccentricity ratio ε for different values of skewness ε∗ . 55

5.1 Exponential Slider Bearing . . . . . . . . . . . . . . . . . . . . . . . . 595.2 DL LSC versus magnetic parameter µ∗ for different values of SD σ∗ . . 655.3 DL LSC versus magnetic parameter µ∗ for different values of skewness

ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 DL LSC versus magnetic parameter µ∗ for different values of mean α∗ 675.5 DL LSC versus magnetic parameter µ∗ for different values of RPP γ . . 685.6 DL LSC versus SD σ∗ for different values of skewness ε∗ . . . . . . . 695.7 DL LSC versus SD σ∗ for different values of mean α∗ . . . . . . . . . 705.8 DL LSC versus skewness ε∗ for different values of mean α∗ . . . . . . 71

6.1 Several Distribution Function . . . . . . . . . . . . . . . . . . . . . . . 766.2 Infinitely Long Slider Bearing . . . . . . . . . . . . . . . . . . . . . . 776.3 DL PD versus RPP α for different values of surface roughness height c∗ 826.4 DL PD versus RPP α for different values of FTR τ . . . . . . . . . . . 836.5 DL PD versus surface roughness height c∗ for different values of RPP α 846.6 DL PD versus FTR τ for different values of RPP α . . . . . . . . . . . 856.7 Coefficient of friction µ versus RPP α for surface roughness height c∗ . 86

xvii

List of Figures xviii

6.8 Coefficient of friction µ versus RPP α for FTR τ . . . . . . . . . . . . 876.9 Coefficient of friction µ versus FTR τ for different values of surface

roughness height c∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

List of Tables

3.1 Variation in DL LSC with respect to mean α∗ for different values of SDσ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Variation in DL LSC with respect to mean α∗ for different values ofskewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Variation in DL LSC with respect to SD σ∗ for different values ofskewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.4 Variation in DL LSC with respect to FTR τ for different values ofskewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of SD σ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.6 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of skewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.7 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of mean α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of mean α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of skewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of SD σ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of eccentricity ratio ε . . . . . . . . . . . . . . . . . . . . . . . 49

4.5 Variation in DL LSC with respect to SD σ∗ for different values of meanα∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50


4.7 Variation in DL LSC with respect to mean α∗ for different values ofskewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of mean α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.9 Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of SD σ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.10 Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of skewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Relation between C, r and H . . . . . . . . . . . . . . . . . . . . . . . 625.2 Variation in DL LSC with respect to magnetic parameter µ∗ for different

values of SD σ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3 Variation in DL LSC with respect to magnetic parameter µ∗ for different

values of skewness ε∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Variation in DL LSC with respect to magnetic parameter µ∗ for different

values of mean α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

xix

List of Tables xx

5.5 Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of RPP γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


5.7 Variation in DL LSC with respect to SD σ∗ for different values of meanα∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.8 Variation in DL LSC with respect to skewness ε∗ for different values ofmean α∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1 Selection of DL parameters . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Variation in DL PD with respect to RPP α for different values of surface

roughness height c∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Variation in DL PD with respect to RPP α for different values of FTR τ 836.4 Variation in DL PD with respect to surface roughness height c∗ for

different values of RPP α . . . . . . . . . . . . . . . . . . . . . . . . . 846.5 Variation in DL PD with respect to FTR τ for different values of RPP α 856.6 Variation in coefficient of friction µ with respect to RPP α for different

values of surface roughness height c∗ . . . . . . . . . . . . . . . . . . . 866.7 Variation in coefficient of friction µ with respect to RPP α for different

values of FTR τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.8 Variation in DL PD with respect to FTR τ for different values of surface

roughness height c∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Dedicated to the almighty.........

xxi

CHAPTER 1

Introduction

1

Introduction 2

1.1 Abstract

The present study is related with numerical modeling based analysis of hydrodynamic

bearings. The numerical modeling of hydrodynamic bearings leads to an ordinary

differential equation with appropriate boundary conditions whose approximate

analytical solutions are achieved in graphical forms.

Tribology is a term that is related to the friction, wear and lubrication of surfaces,

which have relative motion. Tribology is connected to the mechanism of lubrication,

friction control and reducing and elimination of losses owing to friction and wear in

mechanical system. Tribology includes all phenomena that occur on frictional surfaces

and is related closely to various scientific fields such as mechanical engineering,

physics, mathematics, chemistry and material science. For understanding such

phenomena, mathematical model based numerical analysis may be useful to us.

Tribology is important to modern machinery, which utilize sliding and rolling surfaces.

The intention of research in Tribology is understandably the minimization and

elimination of losses resulting from friction and wear at all levels of technology where

the rubbings of surfaces are involved.

The lubricants generally, decrease wear and heat between touching surfaces in relative

motion. Lubricants are found in solid, liquid and gaseous forms. Several methods have

been implemented to reduce the negative effect of surface roughness on the

performance of bearing systems. One such method is the replacement of conventional

lubricants by a magnetic fluid lubricant. The magnetic fluid is a suspension of solid

magnetic particles of sub domain size in liquid carrier. Depending upon the

ferromagnetic material and the method of preparation the mean diameter of a particle

varies from 3 nm to 15 nm. The advantage of magnetic fluid lubricant over the

conventional one is that the magnetic fluid can be maintained at a desired location by

an external magnetic field. The application of magnetic fluid as a lubricant modifying

the efficiency of the bearing system has been an intensive field of investigations and it

is not surprising that the magnetization in variably results to improved bearing

Abstract 3

performance.

Till today, it is a well-known fact that after having some run in and wear the bearing

surfaces creates roughness. The roughness is random in nature without following any

definite structural pattern. It is well defined that the roughness has a negative effect on

the performance of the bearing system. When the proportion of roughness becomes

more, the life span of the bearing system gets drastically decreased. The surface

roughness has been a matter of debate in various recent researches on account of its

negative effect.

The present study is related with the researches of performance of a magnetic fluid

based rough slider bearings and rough short journal bearings.

The content of the thesis is planned in six chapters.

The current-FIRST chapter is a about introduction to the thesis that contains different

sections like abstract of the thesis, brief description on the state of the art of the

research topic, definition of the problem, objective and scope of work, original

contribution by the thesis, methodology of research and results/comparisons,

achievements with respect to objectives, conclusion of the thesis, list of publications.

Chapter 2 deals with the main constituents of Tribology, namely, friction, wear and

lubrication. Different types of lubricants, various types of bearing, geometries of the

surfaces etc. are treated for bearing design characteristics. Also provides the governing

equations for fluid flow in general and magnetic fluid flow in particular. The method of

deriving the associated generalized Reynolds equation for the pressure distribution in a

bearing system is introduced here. This modified Reynolds type equation considers

account the effect of magnetic fluid, geometry of the surfaces and surface roughness.

The mathematical modeling of bearing system is found here.

Introduction 4

Chapter 3 is concerned with the effect of roughness of a magnetic fluid based infinite

long SB. Neuringer Rosenwicg model for the magnetic fluid flow is used here. With a

view to evaluating the TR, the stochastic models given by Christensen and Tonder have

been applied here. The associated Reynolds equation is solved with suitable boundary

conditions and obtains the PD and in sequence we got LSC. The results displayed in

graphical forms suggest that the negative effect of the SD and FTR can be

compensated to the positive effect of the magnetization parameter in the case of

negatively skewed roughness.

Chapter 4 describes the performance of a magnetic fluid based rough hydrodynamic

short JB with TR. Regarding roughness, the method adopted by Christensen and

Tonder finds the application here in statistical averaging of the associated Reynolds

equation. The results displayed in graphical forms show that the eccentricity ratio

plays a key role in improving the bearing performance. The magnetization tries to

compensate the adverse effect of roughness.

Chapter 5 presents the effect of magnetic fluid through a series of flow factors, which

is strongly dependent on the RPP on the behavior of a longitudinally rough exponential

SB. With a view to evaluating the LR, the stochastic models given by Christensen and

Tonder have been applied here. The associated Reynolds equation is done with

appropriate boundary conditions to obtain the PD. From this, the expression for LSC is

achieved. The results displayed in graphical forms show that the magnetization gets

higher the LSC while the LSC gets reduced due to the SD. Moreover, it is observed

that the increment in the positively skewed roughness longitudinally creates the loss in

the LSC of the bearing.

Chapter 6 describes the effect of several probability density functions related with

roughness characteristics on the performance of hydrodynamic SB. The stochastic

theory of Christensen and Tonder has been used to evaluate the effect of three different

types of PDF related with roughness characteristics. The modified Reynolds type

Definition of the problem 5

equation is solved with appropriate boundary conditions to achieve the PD. Moreover,

friction is counted for several roughness models. It is found that the FTR and RPP can

have crucial role in improving the bearing performance. So the performance

characteristics can be made better by selecting design parameter carefully. More over

the bearing performance made better in Beta distribution compere to other

distributions.

The study conducted in this dissertation makes it mandatory that even if suitable

magnetic strength is in place, the roughness aspect must be considered carefully while

designing the bearing system. It is important to note that the bearing can support a load

even in the absence of flow, unlike the traditional lubricants.

1.2 Definition of the problem

The underlying mathematical model for hydrodynamic SB. Here, we considered that

bearing surfaces are rough. Regarding roughness, the method adopted by Christensen

and Tonder finds the application here in statistical averaging of the associated

Reynolds equation. Here, the magnetic fluid is used instead of conventional fluid.

Neuringer Rosenwicg model for the magnetic fluid flow is used here.

The model governing the PD in a SB is the generalized Reynolds’ equation is

introduced by Osborne Reynolds [50].

∂

∂x

(ρh3

η

∂ p∂x

)+

∂

∂ z

(ρh3

η

∂ p∂ z

)= 6U

∂

∂x(ρh) (1.1)

Introduction 6

where,

ρ is density of fluid

η is viscosity of fluid

h is fluid film thickness

p is pressure of fluid film

U is Velocity of bearing surfaces

1.3 Objective and Scope of work

The main purpose for this study is to understand the combined effect of surface

roughness and magnetization on performance of the SB.

Moreover, it is aimed to analyze the behavior of present mathematical model in terms

of PD, friction coefficient and LSC in presence of ferro-lubricant in compared to the

conventional lubricant.

Apart from this, our aim in the last chapter is to compare the performance of the SB

system through three different types of model for roughness.

The studies included in this thesis suggest that the investigations can be modified and

developed to work in the directions given below:

• The effect of surface roughness may be studied for various kinds of magnetic fluid

based bearings using the magnetic fluid flow model of Shliomis and Jenkins.

• Here, the results are gained for one dimensional mathematical model that can be

extended for two dimensional models.

Original contribution by the thesis 7

1.4 Original contribution by the thesis

The original contribution by the thesis is mathematical modeling on hydrodynamic SB,

which analyzes:

• The effect of roughness parameters on the performance of magnetic fluid based

infinitely long SB.

• The effect of magnetic fluid as a lubricant equipped with the roughness

parameters on the performance of short JB.

• The effect of several probability density functions related with roughness

characteristics on the performance of hydrodynamic SB.

1.5 Methodology of Research and Results/Comparisons

The following assumptions were considered in the model:

• Body forces are avoided.

• The lubricant is assumed as Newtonian

• The viscosity is constant throughout film thickness.

• The flow is assumed to be steady in X- direction and the surface roughness is

assumed either longitudinal or transverse.

The problem is dealt as a one dimensional problem. Several parameters of roughness,

like mean, SD and skewness, RPP for LR and TR and magnetization parameters are

introduced at different stages and then solved the one dimensional differential equation

for the PD at the contact zone of the bearing system with appropriate boundary

Introduction 8

conditions. The LSC and the friction coefficient are obtained.

The integrals occurring in the calculation throughout the work is carried out by

Simpson’s 1/3-rule. And the results and mutual relations between two parameters are

shown graphically and in tabular form. Various roughness models are taken in account

for comparing the corresponding PD, LSC and coefficient of friction.

1.6 Achievements with respect to objectives

The generalized Reynolds’ equation is modified accordingly to achieve our goal and :

• Solved the modified mathematical model with respect to suitable boundary

conditions for getting relation among various parameters like roughness

parameters (e.g. mean, SD, skewness) , pattern of roughness (e.g. TR or LR),

type of lubricant (e.g. magnetic lubricant or conventional lubricant), magnetic

parameter, shape of bearing geometry etc.

• Achieved satisfactory results as desired and obtained suitable combinations of

such kind of parameters that may enhance the performance and life period of the

bearing system.

1.7 Conclusion

• Magnetic fluid as a lubricant enhances the bearing performance for Neuringer-

Rosensweig magnetic flow model for SB.

• It is found from the present study that magnetic fluid based bearing system

supports certain amount of load, even in the absence of flow irrespective of

roughness which is very unlikely, in the case of traditional lubricant based

bearing system.

Conclusion 9

• However, negatively skewed roughness remains beneficial from design point of

view, when the surfaces are transversely rough.

• It is observed that magnetic fluid may move to some extent in mitigating the

negative effect of roughness.

CHAPTER 2

A Prerequisite for Tribology

10

Friction 11

Tribology is a term that is concerned to the friction, wear and lubrication of surfaces

having relative motion. The subject tribology is related to the mechanism of

lubrication, friction control and reducing and elimination of losses due to friction and

wear in mechanical system. Most of mechanical system is based on the tribology. The

performance and durably in mechanical system can be achieved with the help of

tribology. The significance of tribology has increased with the passage of time. The

progress of human being is dependent on the practices of tribology.

2.1 Friction

The resistance to motion experienced during sliding or rolling is called friction. It is

created when two material elements slide against each other. Friction is a system

response. High friction is created when two solid surfaces come in contact without any

lubricant.

Up to some extent it can be said that friction is a boon to mankind, since the movement

and transportation is possible only because of it. Even in some mechanical applications

like clutches and brakes, friction is maximized. Apart from this, friction is not

expected in mechanical components like bearings and seals due to its resistance in

performance. Friction is required to be minimized in such cases.

The different types of frictions are:

2.1.1 Dry Friction

It is also known as coulomb friction. It happens when two dry surfaces move to one

another. It is sub divided into two types of frictions.

• Static friction

• Sliding friction

A Prerequisite for Tribology 12

Static friction occurs between non-moving surfaces and sliding occurs when two objects

are moving.

2.1.2 Fluid friction

The friction that is created when two layers of viscous fluid are moving relatively to

each other.

2.1.3 Lubricated Friction

When a fluid separates two solid surfaces, it is called lubricated friction.

2.1.4 Skin friction

The component of drag, the force that resists the motion of a fluid across the surface of

a body.

2.2 Wear

Wear occur due to friction. It is the damage or removal of material from one or

both of two solid surface in sliding, rolling or impact motion relative to one another.

Wear occurs though surface interactions at asperities the actual loss of material are

preceded by wear damage. High-speed engines of modern generations which can cause

operational disturbance or destructions of engines.

2.3 Lubrication

The science that studies the reduction of friction offer applying the substance called

lubricant is called lubrication.

Lubrication 13

The lubrication aims at decreasing the energy loss and wear by reducing the friction

between two surfaces in relative motion through the presence of film of lubricant.

Lubrication plays a major role in carrying away the heat generation due to friction and

protecting the bearings from corrosion. The types of lubricating based on the thickness

of film of lubricant are as follows.

2.3.1 Hydrodynamics Lubrication

Two surfaces moving at some relative velocity with respect to each other are separated

by a fluid film. In this process the pressure is generated by virtue of relative motion only.

The design of bearings may wear when started, stopped or reversed, as the lubricant

break down. One of the drawbacks that is found in this is that loads can not carried at

low speed and an appreciable wear is also found due to frequent startup and stop.

2.3.2 Hydrostatic Lubrication

Hydrostatic bearings are those working under the hydrostatic mode. In this process the

fluid lubricated film is maintained by the application of external force, so that it is not

squeezed out and remain where it is. The predetermined performance characteristics

such as flow, load capacity, stiffness, friction and pumping power. Mostly the low

coefficient of friction and extremely high stifles are given by the hydrostatic bearings.

Several studies on hydrostatic bearings have been found

2.3.3 Boundary Lubrication

When complete separation of the moving surface by a lubricant film can not be

maintained, the kind of lubrication, which occurs is called boundary lubrication. When

the condition of bearing design, speed, load and method of application of lubricant do

not allow the formation of a separating lubricant film by hydrodynamic action,

boundary lubrication is the state of lubrication exists.


2.3.4 Mixed Lubrication

It occurs in a situation between the hydrodynamic and boundary lubrication. In such

lubrication, the fluid film between the two sliding surface is thin and the surface

asperities begin to interfere in hydrodynamic process. Asperity contact and fluid film

are supported partly by the load.

2.3.5 Elasto-hydrodynamic Lubrication

If the pressure is high enough, it can distort the bearing on slider on both and in doing

so change the pressure distribution. The study of this effect is particularly called

elasto- hydrodynamic lubrication In the lubrication of gear and roller bearings where

high pressure can be developed, this effect is particularly important.

2.4 Bearings

The system of machine elements which function to support an applied load by

reducing friction between the relatively moving surfaces is called a bearing. The

separation of these surfaces cab be done by a lubricant film, which can be liquid,

semisolid or gas.

The direction of applied load classifies the bearings. The redial or journal bearing

supports a redial load. The thrust or an axial load is supported by a thrust bearing. The

conical bearings supports both redial and axial loads.

Mainly two type of bearings are used in practice. They are rolling elements and fluid

film bearings. Since the rolling friction is lower then the sliding friction, the rolling

bearing elements are used widely in industry.

Bearings 15

2.4.1 Hydrodynamic Bearings

As shown in Fig.2.1 hydrodynamic bearing consist of two solid surface separated by

fluid film and slightly inclined towards one another. It develops positive pressure by

virtue of relative motion of two surfaces separated by a fluid film. They differ in both

their size and in the load they support.

One of most frequent function of hydrodynamic SB is to support rotating shafts when

the load vector and the axis of rotation are parallel.

The hydrodynamic JB is employed when the motion is rotational and the load vector is

perpendicular to the axis of rotation.

FIGURE 2.1: Slider Bearing and Journal Bearing

2.4.2 Hydrostatic Bearings

Hydrostatic bearings work under the hydrostatic lubrication mode. The fluid lubricated

film is maintained by the application of external force, so that it is not squeezed out

and remains where it is.


FIGURE 2.2: Hydrostatic Bearing

2.5 Tribological Surface

On a microscopic scale all surfaces are rough. No solid surface is perfectly smooth on

atomic scale. The absolute height of asperities and valleys vary greatly the effect of

surface roughness plays a major role in tribology. When two solid surfaces come in

close proximity actual contact in made only by the asperities of the two surfaces. The

real area of contact, which is the totality of the individual asperity contact area is only a

fraction of the apparent area of contact.

2.6 Lubricant

The substance that reduce friction between two surfaces in mutual contact. It decreases

the generation of heat during the movement of the surface. The use of lubricant depends

on the type of bearings used in the machine elements. The different types of lubricants

are solid, gases even plasma are used as lubricants. The dry rubbing bearing use solid

lubricants. Due to contamination reasons, lubricating oils and gases can not be used

in such cases dry lubricants are used. Grease and polymer-thickened oils are common

semi solids lubricants. Mostly vegetable oil, minerals or petroleum oil, synthetic oil are

used as liquid lubricants.

Extension of the Classical Theory 17

2.7 Extension of the Classical Theory

The lubrication characteristics of a bearing system is based upon the following:

• Nature of lubricant: Liquid or gas, Newtonian or Non-Newtonian behavior,

compressible or incompressible lubricant, pressure may change along the fluid

film thickness, variation of viscosity

• Nature of surface: Elasticity, thermal conductivity, hardness, surface roughness

etc.

• Effect of flow regimes: Bearing geometry, thickness of fluid film, boundary

conditions etc.

We can study the behavior of any lubricated system by using a mathematical model

which is formed considering the factors given above which is based on the given

physical situation. The bearing characteristics such as LSC, friction forces, coefficient

of friction etc. based upon the lubrication process and the pressure generated in the

fluid film.

In this thesis, the surface roughness effect on different characteristics of SB with help

of several probability density functions and the combined surfaces roughness and

magnetization effect on SB as well as JB will be investigated.

2.7.1 Basic Equation

The Reynolds equation is the base for the theoretical study of hydrodynamic

lubrication, which is derived by Reynolds [50]. The formation of a thin lubricant film

assumed to be the basic mechanism of hydrodynamic lubrication is shown by the

classical experiment by Beauchamp Tower. Tower conclusions were explained by

Reynolds through his equation. Navier-Stokes equation is the base for Reynods


equation, which contains density and viscosity term as parameters.

The Navier-Stokes equation and equation of continuity in their most general form a

Newtonian fluid can be written as in following form,

ρDudt

=ρX− ∂ p∂x

+∂

∂y

(η

(∂u∂y

+∂v∂x

))+

∂

∂ z

(η

(∂w∂x

+∂u∂ z

))+

∂

∂x

(η

(2

∂u∂x− 2

3

(∂u∂x

+∂v∂y

+∂w∂ z

)))(2.1)

ρDvdt

=ρY − ∂ p∂y

+∂

∂ z

(η

(∂v∂ z

+∂w∂y

))+

∂

∂x

(η

(∂u∂y

+∂v∂x

))+

∂

∂y

(η

(2

∂v∂y− 2

3

(∂u∂x

+∂v∂y

+∂w∂ z

)))(2.2)

ρDwdt

=ρZ− ∂ p∂ z

+∂

∂x

(η

(∂w∂x

+∂u∂ z

))+

∂

∂y

(η

(∂v∂ z

+∂w∂y

))+

∂

∂ z

(η

(2

∂w∂ z− 2

3

(∂u∂x

+∂v∂y

+∂w∂ z

)))(2.3)

∂ρ

∂ t+

∂ (ρu)∂x

+∂ (ρv)

∂y+

∂ (ρw)∂ z

= 0 (2.4)

where,

velocity components are u,v and w in x,y and z directions, p is PD, ρ is density and η

is viscosity of the fluid.

Reynolds derived the the generalized Reynolds equation for incompressible fluid from

the Navier-Stokes equation and equation of continuity under some basic assumptions.

The basic assumptions given by Reynolds are as follows:

Extension of the Classical Theory 19

• Body and inertia forces are negligible.

• Across the fluid film, pressure variation is zero.

• No slip at the bearing surfaces.

• No external forces act on the fluid film.

• The fluid flow is laminar

• Velocity gradients in all but y direction are negligible.

• The bearing length is very large compared to the fluid film height.

Using the above the basic assumptions, equations (2.1 to 2.3) can be reduced to

∂ p∂x

= η∂ 2u∂y2 (2.5)

∂ p∂ z

= η∂ 2w∂y2 (2.6)

Integrating equations (2.5) and (2.6) with respect to y two times with the following

boundary conditions:

u = ub,w = wb at y = 0 and u = ua, w = wa at y = h, substituting the results on previous

for u and w so obtained, into the equation of continuity(2.4) and integrating it with the

boundary condition v = vb at y = 0 and v = va at y = h gives the generalized Reynolds

equation:

∂

∂x

(ρh3

η

∂ p∂x

)+

∂

∂ z

(ρh3

η

∂ p∂ z

)= 6U

∂

∂x(ρh) (2.7)


where,

U =ua +ub

2

2.8 Slider bearing

Owing to their stability durability and high LSC, SB are mostly utilized in engineering.

The SB that is considered idealization of single sector shaped pad of a hydrodynamic

thrust bearing, consists of a fixed pad and a moving pad.

2.8.1 Infinitely long slider bearing

FIGURE 2.3: Infinitely Long Slider Bearing

An infinitely long SB is made of two surfaces, which are separated by a fluid film. The

lower plate (slider) moves with the uniform speed U . The bearing geometry is displed

in Fig. 2.3. Here we assume that it is infinitely long in z direction. So there is no

variation in pressure in z direction. So the∂ p∂ z

term in equation (2.7) can be neglected.

Finally, the form of Reynolds type equation for infinitely long slider bearing is

Journal bearing 21

ddx

(ρh3

η

d pdx

)= 6U

ddx

(ρh) (2.8)

2.9 Journal bearing

A JB are used to carry radial load. A simple JB made of of two cylinders. The outer

cylinder is called bearing and inner rotating shaft. The journal bearings are used in

many industrial applications.

2.9.1 Infinitely short journal bearing

FIGURE 2.4: Infinitely Short Journal Bearing

The bearing geometry displayed in Fig. 2.4. R j and Rb are radius of journal and bearing

respectively. The journal rotates inside a bearing with uniform velocity U . The bearing

is infinitely short in Z-direction. So there is no variation in pressure in x direction,

therefore the∂ p∂x

term in equation (2.7) can be neglected.

The Reynolds equation for PD in this case will be


ddz

(ρh3

η

d pdz

)= 6U

ddx

(ρh) (2.9)

Finally, the governing Reynolds equation in polar coordinates will be

ddz

(ρh3

η

d pdz

)=

6UR j

ddθ

(ρh) (2.10)

2.10 Magnetic fluid lubrication

In the presence of a magnetic field, the magnetic fluid is magnetized strongly. Three

components are necessary to make ferrofluid like magnetic particles of colloidal size,

carrier liquid and stabilizer.

The supply of lubricating medium only to the friction zone in possible through the

magnetic fluid, and it can be positioned in this zone with the help of magnetic fluid in a

specific design. The infinitely long JB with axial magnetic field have been studied by

Elco and Huges [17]. The problem of lubrication under the influence of a uniform

magnetic field is discussed by Agrawal [1]. They have concluded that the LSC if

bearing was increased by the application of magnetic field. The analysis of an

electrically conducting lubricant based infinitely long JB was presented by Kuzma

[23].

The performance of a magnetic fluid based squeeze film was studied Verma [60]. The

conclusion of the studies is that a positive effect is created on the bearing system with

the help of magnitization. The analysis of verma was further extended by Bhat and

Deheri [5] by analyzing the performance of ferro fluid lubricated porous annular disks.

The theoretical investigation of ferro fluid based hydrodynamic JB with the help of

current carrying wire model was analyzed by Nada et al. [35]. The theoretical analysis

of the PD in a ferro fluid based hydrodynamics JB is conducted by Urreta et.al. [58].

Magnetic fluid lubrication 23

2.10.1 Neuringer-Rosensweig Model

Neuringer and Rosenweig [32] studied an analytical solution for the problem of source

flow with heat addition with a view to displaying the thermo magnetic and

magneto-mechanical effect attendant to simultaneous heat addition and fluid motion.

The model consisted of the following equations:

ρ(q.∇)q =−∇p+η∇2q+µ0(M.∇)H (2.11)

∇.q = 0 (2.12)

∇×H = 0 (2.13)

M = µH (2.14)

∇.(H +M) = 0 (2.15)

Using equation (2.11) and equation (2.12), equation (2.9) turns out to be

ρ(q.∇)q =−∇

(p− µ0µM2

2

)+η∇

2q (2.16)


We can see here, the extra pressureµ0µM2

2is created into the Navier-Stockes equation

when ferro fluid is used as a lubricant.

2.10.2 Modified Reynolds equation of infinitely long slider bearing

for magnetic fluid

Thus, the modified Reynolds equation in this case is obtained as

ddx

(ρh3

η

ddx

(p− µ0µM2

2)

)= 6U

ddx

(ρh) (2.17)

2.10.3 Modified Reynolds equation of infinitely short journal

bearing for magnetic fluid

ddz

(ρh3

η

ddz

(p− µ0µM2

2)

)=

6UR j

ddθ

(ρh) (2.18)

2.11 Surface Roughness Effect

Earlier it was assumed that the bearing surfaces are perfectly smooth. Now it is clear

that this assumption is unrealistic. A lot of research has been done on surface

roughness effect on hydrodynamic lubrication because of the fact that all surfaces are

rough on micro scale. The effect of surface roughness on the hydrodynamic lubrication

of bearings has been studied by researchers through various approaches. The effect of

surface roughness is taken into account by considering that the film thickness is a

function of surface roughness.

Surface Roughness Effect 25

It was recognized by Halton in 1958 that it was not realistic to believe that smooth

mathematical planes can represent the bearing surfaces while in working with small

film thickness. Burton [9] examining the Reynolds equation and also presented a more

realistic representation Bearing. The Reynolds equation was modified with stochastic

concepts related to surface roughness by Tzeng and Saible [57] and they have been

successful in carrying through an analysis of inclined SB.

It was again refined by Christensen and Tonder [11] by proposing the stochastical

analysis with a view to evaluating the effect of surface roughness. The type of

roughness decides the increment and decrement of bearing performance was the

conclusion by them. In this stochastic model lubricant film thickness H is considered

as

H = h(x)+δ

where, h(x) is the lubricant mean film thickness and δ is the deviation from h(x). The

δ is taken to be stochastic in nature and described by the probability density function

f (δ ),−c≤ δ ≤ c

where c is the maximum deviation from the lubricant mean film thickness. The mean

α , the SD σ and skewness ε associated with random variable δ are governed by the

relations

α = E(δ )

σ2 = E[(δ −α)2

]

and


ε = E[(δ −α)3

]

E denotes the expected value given by

E () =∫ c−c () f (δ )dδ

Christensen at. al [12] also derived a generalized Reynolds equation applicable to

rough surfaces by assuming that the film thickness function is a stochastic process.

The application of flow model to lubrication between rough sliding surfaces has been

studied by Patir and Chang [45]. The roughness effect on the performance of infinitely

long and narrow porous bearings has been analyzed by Gururajan and Prakash [20]

[21].

CHAPTER 3

Magnetic Fluid Based an Infinitely

Long Transversely Rough Slider

Bearing

27

Magnetic Fluid Based an Infinitely Long Transversely Rough Slider Bearing 28

3.1 Introduction

A crucial importance has been allotted to fluid with strong magnetic properties for the

last decade. Now a days the magnetic fluid in bearing system has attracted in

mechanical field a lot. Many mechanical instruments have been made by the

application of magnetic fluid. The significant aspect of magnetic fluid is that we can

sustain it at particular location according to the requirement.

Earlier the with a view to improving the performance of the bearing system, magnetic

fluid is getting more and more applicable. Many researchers like ( Nada and Osman

[29], Agrawal[2], Huang et al.[22], Bhat and Deheri [7], Odenbach [33], Urreta et

al.[59] ) have investigated the hydrodynamic bearings under the influence of magnetic

fluid. It is found that the bearing system function more efficiently in presence of

magnetic fluid.

SB are mostly utilized because of their stability, durability and high LSC in

engineering field. The surface roughness and lubricant properties both are important

factors in hydrodynamic lubrication. Many fundamental problems such as friction,

LSC and heat have the significance of surface roughness evaluation. Because of this,

the theoretical investigations have used surface as the subject for a long time. The

researchers like Tzeng and Saibel [57], Christensen and Tonder [13] [14] [11], Gupta

and Deheri [19] proposed the stochastic approach for evaluate the effect of roughness

and presented more realistic bearing surface. The effect of TR on the behavior of thin

film lubrication at nano scale of a slider bearing is studied by Patel and Deheri [37].

The combined effect of thermal and roughness on an infinite tilted pad SB was

discussed by Sinha and Adamu [56]. The performance of a magnetic fluid based

hydrodynamic curved annular plates was studied by Lin et al. [26]. The effect of TR

on a SB with ferro fluid was analyzed by Deheri et al. [16]. The performance of a

magneto-hydrodynamic squeeze film formed parallel annular plates was observed by

Lin et al. [28]. The performance of a rough short bearing incorporating deformation

effect under magnetic fluid lubrication was studied by Shimpi and Deheri [53]. Deheri

et al.[41] investigated the performance of a ferro fluid based hydrodynamic long JB.

The performance of a rough porous hyperbolic SB with ferro fluid was discussed by

Analysis 29

Deheri et al. [43].

Here, it has been proposed to discuss the effect of roughness on the performance of a

magnetic fluid lubricated infinitely long SB.

3.2 Analysis

The bearing geometry is given in Fig. 3.1. Here we assume that SB is infinitely long in

z direction. The lower plate (slider) moves with the uniform speed U .


The stochastic lubricant film thickness is considered as suggested by Christensen and

Tonder [13] [11] [14] in the form:

H = h(x)+δ (3.1)

where, h(x) is the lubricant mean film thickness and δ is the deviation from h(x).

The mean α , the SD σ and skewness ε associated with random variable δ are


governed by the relations

α = E(δ )

σ2 = E[(δ −α)2

]

and

ε = E[(δ −α)3

]

E denotes the expected value given by

E () =∫ c−c () f (δ )dδ

In present study, ferro fluid is taken as lubricant. Following the investigations of Bhat

[4] the magnitude M of the magnetic field H is taken as:

M2 = Kx(l− x) (3.2)

Making use of stochastic modelling of Christensen and Tonder[13] [11] [14], and Sinha

[56], Agrawal [2] and Deheri [41], the form of modified Reynolds equation governing

the PD for magnetic fluid lubricated infinitely long transversely rough SD under the

usual assumptions of hydrodynamic lubrications is:

ddx

(1η

E(H3)ddx

(p− µ0µM2

2

))= 6U

dhdx

(3.3)

Analysis 31

Where,

E(H3) = h3 +3hσ2 +3h2α +3hα2 +3σ2α +α3 + ε

The relevant boundary conditions are:

p = 0 at x = 0 and x = l

Introducing the DL quantities

h∗ =hh0

, α∗ =α

h0, σ∗ =

σ

h0, ε∗ =

ε

h30, G(H) =

E(H3)

h30

,X =xl, P =

ph20

6Uη l, µ∗ =

µ0µKlh20

6ηU

the equation (3.3) reduces to

ddx

(ddx

(P+

12

µ∗X(X−1)

)G(H)

)=

dh∗

dx(3.4)

Integrating both sides one finds that

ddx

(P+

12

µ∗X(X−1)

)=

h∗−h∗mG(H)

(3.5)

where h∗m is the film thickness at maximum pressure


In view of the boundary conditions

P = 0 at X = 0 and X = 1

Then, the expression for DL PD is found to be:

P(X) =12

µ∗X(1−X)+

∫ X

0

h∗−h∗mG(H)

dX (3.6)

where,

h∗m =

∫ 10

1G(H)

dX

∫ 10

h∗

G(H)dX

Lastly, the DL LSC is obtained from the relation:

W =∫ 1

0P(X)dX (3.7)

Results and discussions 33

3.3 Results and discussions

It is clearly visible that the equation:

P(X) =12

µ∗X(1−X)+

∫ X

0

h∗−h∗mG(H)

dX (3.8)

determines the DL PD while the DL LSC is obtained from:

W =∫ 1

0P(X)dX (3.9)

Both these equations depend on various parameters such as µ∗, α∗, σ∗, ε∗. The effect

of magnetization is indicated by µ∗, where as the effect of roughness is determined by

the rest of three parameters.


FIGURE 3.2: DL LSC versus mean α∗ for different values of SD σ∗

TABLE 3.1: Variation in DL LSC with respect to mean α∗ for different values of SDσ∗

σ∗ α∗ =−0.05 α∗ =−0.025 α∗ = 0 α∗ = 0.025 α∗ = 0.05

0 0.198267309 0.19697254 0.195728985 0.194534039 0.193385265

0.05 0.198215705 0.19692373 0.195682827 0.194490393 0.193343993

0.1 0.198061181 0.19677764 0.195544722 0.194359836 0.19322056

0.15 0.197804761 0.196535397 0.195315852 0.194143567 0.193016153

0.2 0.197448571 0.196199183 0.194998397 0.193843727 0.192732851


FIGURE 3.3: DL LSC versus mean α∗ for different values of skewness ε∗

TABLE 3.2: Variation in DL LSC with respect to mean α∗ for different values ofskewness ε∗

ε∗ α∗ =−0.05 α∗ =−0.025 α∗ = 0 α∗ = 0.025 α∗ = 0.05

−0.05 0.198166162 0.196879829 0.195643752 0.194455463 0.193312636

−0.025 0.198114498 0.196829384 0.195594748 0.194408051 0.193266913

0 0.198061181 0.19677764 0.195544722 0.194359836 0.19322056

0.025 0.198006373 0.19672472 0.195493769 0.194310891 0.193173631

0.05 0.197950223 0.196670737 0.195441976 0.194261281 0.193126178

Fig. 3.2 indicates that DL LSC decrease with increased SD. Moreover, mean (+Ve)

decreases the LSC while the LSC increases due to to mean (-Ve). Similar is the trends

of skewness as far as LSC is concerned. (Fig.3.3))


FIGURE 3.4: DL LSC versus SD σ∗ for different values of skewness ε∗

TABLE 3.3: Variation in DL LSC with respect to SD σ∗ for different values ofskewness ε∗

ε∗ σ∗ = 0 σ∗ = 0.05 σ∗ = 0.1 σ∗ = 0.15 σ∗ = 0.2

−0.05 0.195826813 0.195780978 0.195643752 0.195416075 0.195099801

−0.025 0.195778467 0.195732456 0.195594748 0.195366411 0.195049471

0 0.195728985 0.195682827 0.195544722 0.195315852 0.194998397

0.025 0.195678471 0.195632194 0.195493769 0.195264481 0.194946649

0.05 0.195627021 0.19558065 0.195441976 0.195212375 0.19489429


FIGURE 3.5: DL LSC versus FTR τ for different values of skewness ε∗

TABLE 3.4: Variation in DL LSC with respect to FTR τ for different values ofskewness ε∗

ε∗ τ = 3.7 τ = 3.75 τ = 3.8 τ = 3.85 τ = 3.9

−0.05 0.069115149 0.068370405 0.067643752 0.066934693 0.066242738

−0.025 0.069060447 0.068318632 0.067594748 0.066888306 0.066198828

0 0.06900474 0.068265845 0.067544722 0.066840893 0.066153886

0.025 0.068948125 0.068212139 0.067493769 0.066792545 0.066108006

0.05 0.06889069 0.068157603 0.067441976 0.066743349 0.06606127

The message of Fig. 3.4 is that the combined effect of the SD and positively skewed

roughness decreases the LSC. Fig. 3.5 suggests that DL LSC decrease with increased

FTR.


TABLE 3.5: Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of SD σ∗

σ∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

0.0 0.051062318 0.059395651 0.067728985 0.076062318 0.084395651

0.05 0.05101616 0.059349493 0.067682827 0.07601616 0.084349493

0.1 0.050878055 0.059211388 0.067544722 0.07601616 0.084211388

0.15 0.050649185 0.058982518 0.067315852 0.075649185 0.083982518

0.2 0.050331731 0.058665064 0.066998397 0.075331731 0.083665064

TABLE 3.6: Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of skewness ε∗

ε∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

−0.05 0.050977085 0.059310419 0.067643752 0.075977085 0.084310419

−0.025 0.050928081 0.059261414 0.067594748 0.075928081 0.084261414

0.0 0.050878055 0.059211388 0.067544722 0.075878055 0.084211388

0.025 0.050827103 0.059160436 0.067493769 0.075827103 0.084160436

0.05 0.050775309 0.059108643 0.067441976 0.075775309 0.084108643

Conclusion 39

TABLE 3.7: Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of mean α∗

α∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

−0.05 0.096061181 0.147061181 0.198061181 0.249061181 0.300061181

−0.025 0.09477764 0.14577764 0.19677764 0.24777764 0.29877764

0.0 0.093544722 0.144544722 0.195544722 0.246544722 0.297544722

0.025 0.092359836 0.143359836 0.194359836 0.245359836 0.29635986

0.05 0.09122056 0.14222056 0.19322056 0.24422056 0.29522056

Tables 3.5 to 3.7 represent the variation of DL LSC with respect to µ∗ for different

values of σ∗, ε∗ and α∗. It is clearly observed that the effect of magnetization on LSC

is positive for all rough parameter.

3.4 Conclusion

In the present investigation, the negative effect of the SD and FTR can be compensated

to the positive effect of the magnetization parameter in the case of negatively skewed

roughness. Further, when we design the bearing system, it is required to evaluate the

roughness aspect.

CHAPTER 4

Magnetic Fluid Based a Short

Transversely Rough Journal Bearing

40

Introduction 41

4.1 Introduction

With the passage of time the researches on performance characteristics of JB with

various shapes and various lubricants have been done by the many authors. The couple

stress effect on squeeze film formed finite JB was investigated by Lin [24]. It was

observed that couple stress provided higher LSC. The theoretical investigation of ferro

fluid based hydrodynamic JB with the help of current carrying wire model was

analyzed by Nada et al. [35]. The result was that the magnetic fluid is more effective

with respect to conventional fluid. The analysis of an electrically conducting fluid

based infinitely long JB was presented by Kuzma [23]. The performance of ferro fluid

based squeeze film formed long JB using the several flow models was discussed by

Shah and Bhat [51]. The couple stress effect in a finite hydrodynamic JB lubricated

with magnetic fluids was investigated by Nada and Osman [29]. The theoretical

investigation of of hydrodynamic short JB with ferro fluid was discussed Deheri et al.

[39]. In all above studies, it is clearly seen that the performance of JB is improved by

the application of magnetic fluid.

The bearing performance may not be assumed by the smooth surfaces of bearing. The

roughness of bearing surfaces is evaluated experimentally and theoretically because it

plays a major role on the performance of bearing system. Many researcher Tzeng and

Saibel [57] Christensen and Tonder[11] have used the stochastic approach to evaluate

the effect of roughness in hydrodynamic bearing system theoretically. The effect of TR

on the short JB under dynamic loading was studied by Raj and Sinha [49]. The effect

of the roughness on the micropolar fluid based short JB under the consideration of

static and dynamic behavior of squeeze film was analyzed by Naduvinamani et al.

[30]. It was found that the journal centre velocity reduced due to micropolar fluid

under a cyclic load. Further, LSC increases and journal centre velocity decrease due to

negatively skewed roughness. The theoretical investigation of a long partial rough JB

under the consideration of oscillating squeeze film behavior was discussed by Lin et

al.[25]. Nanduvinamani and Kashinath [31] Studied the performance characteristics of

squeeze film based rough short JB with micropolar fluid. The effect of surface

roughness on porous infinitely short JB with magnetic fluid was presented by Shimpi

Magnetic Fluid Based a Short Transversely Rough Journal Bearing 42

and Deheri [54]. The hydrodynamic finite JB with magnetic fluid was analyzed by

Deheri et al. [40].

In this study, it has been discussed that the performance of a ferro fluid formed a

transversely rough short JB.

4.2 Analysis

The geometry of the short JB is presented in Fig.(4 1). R j and Rb are radius of journal

and bearing respectively. The journal rotates inside a bearing with uniform velocity U .

FIGURE 4.1: Short Journal Bearing

Following Agrawal [2] the magnitude of the magnetic field is considered as

M2 = K(

z− B2

)(z+

B2

)(4.1)

Making use of a stochastic averaging method of Christensen and Tonder [13] [11] [14]

and Patel [39], Agrawal[2], Bhat [4], Deheri [41], the form of modified Reynolds

Analysis 43

equation for magnetic fluid lubricated infinitely short transversely rough JB is

d2

dz2

(p− µ0µM2

2

)=

6ηUE(H3)R j

dhdθ

(4.2)

where,

E(H3) = h3 +3hσ2 +3hα2 +3h2α +3σ2α +α3 + ε

h = c(1+ εcosθ)

c is Radial clearence

e is Distance between Ob and O j

ε =ec

is Eccentricity ratio

The associated boundary conditions are:

p = 0 at z =±B2

andd pdz

= 0 at z = 0

Introduction of DL quantities

Z =zB, h∗ =

hc, σ∗ =

σ

c, α∗ =

α

c, ε∗ =

ε

c3 , G(H) =E(H3)

c3 , P =R j

Uηp, µ∗ =−

µ0µKR jB2

ηU

leads to the DL Reynolds equation:

d2

dZ2

(P− µ∗

2

(14−Z2

))=−6εsinθ

G(H)

(Bc

)2

(4.3)



P = 0 at Z =±12

anddPdZ

= 0 at Z = 0


P =

(µ∗

2+3(

Bc

)2(εsinθ

G(H)

))(14−Z2

)(4.4)

The LSC in x direction is given by

wx =−2∫

π

0

∫ B2

0pcosθR jdθdz (4.5)

Thus, the DL LSC in x direction is obtained from:

Wx =c2

ηUB3 wx =−14

∫π

0

εsin2θ

G(H)dθ (4.6)

The LSC in z direction is obtained by


wz = 2∫

π

0

∫ B2

0psinθR jdθdz (4.7)

Thus, the DL LSC in z direction is obtained by

Wz =c2wz

ηUB3 =12

∫π

0

εsin2θ

G(H)dθ +

µ∗

6(4.8)

Consequently, the resultant DL LSC is given by

W =√

W 2x +W 2

z (4.9)

4.3 Results and discussions

It is observed that DL PD is obtained from Eq.(4.4) while Eq.(4.9) presents the the DL

LSC. Above both expressions depend on different parameters like mean α∗, SD σ∗,

skewness ε∗ and magnetization parameter µ∗. Removing the all roughness parameters

the study reduces to the performance of hydrodynamic short JB with magnetic fluid as

analyzed by Deheri et al.[39]. From Eq.(4.8), it is cleary seen that DL LSC increases

byµ∗

6due to the magnetization parameter. Figs.(4.2 to 4.5) indicate that LSC increases

with respect to the magnetization parameter.


FIGURE 4.2: DL LSC versus magnetic parameter µ∗ for different values of mean α∗


α∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

−0.05 0.556761926 0.571382266 0.586111851 0.600942646 0.615867342

−0.025 0.504062562 0.518804892 0.533660493 0.548620163 0.563675615

0.0 0.459334928 0.474185797 0.489153677 0.504228148 0.519399929

0.025 0.420877366 0.435826103 0.450895313 0.46607331 0.481349802

0.05 0.387452815 0.402490771 0.417652383 0.43292466 0.448296291


FIGURE 4.3: DL LSC versus magnetic parameter µ∗ for different values of skewnessε∗


ε∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

−0.05 0.51519734 0.529723942 0.544373185 0.559135432 0.574001962

−0.025 0.485099182 0.499800482 0.514621592 0.529552451 0.544584035

0.0 0.459334928 0.474185797 0.489153677 0.504228148 0.519399929

0.025 0.436932473 0.451913125 0.467008046 0.482206504 0.497499011

0.05 0.417205102 0.432299614 0.447505722 0.462812426 0.478210067


FIGURE 4.4: DL LSC versus magnetic parameter µ∗ for different values of SD σ∗


σ∗ µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

0 0.481809271 0.496564711 0.511437589 0.526417949 0.541496872

0.05 0.475948845 0.490729145 0.50562678 0.520631677 0.535734824

0.1 0.459334928 0.474185797 0.489153677 0.504228148 0.519399929

0.15 0.434467913 0.449424615 0.464497747 0.479676334 0.494950674

0.2 0.404471927 0.419556504 0.434756641 0.450060628 0.465458222


FIGURE 4.5: DL LSC versus magnetic parameter µ∗ for different values of eccentricityratio ε

TABLE 4.4: Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of eccentricity ratio ε

ε µ∗ = 0.1 µ∗ = 0.2 µ∗ = 0.3 µ∗ = 0.4 µ∗ = 0.5

0.1 0.09452708 0.111118679 0.127729826 0.144353772 0.160986553

0.2 0.183053015 0.199327359 0.215661637 0.232043191 0.248462672

0.3 0.296712786 0.312405421 0.328194107 0.344065622 0.360009012

0.4 0.459334928 0.474185797 0.489153677 0.504228148 0.519399929

0.5 0.716935812 0.730698153 0.744579195 0.758572422 0.77267174


FIGURE 4.6: DL LSC versus SD σ∗for different values of mean α∗

TABLE 4.5: Variation in DL LSC with respect to SD σ∗ for different values of meanα∗

α∗ σ∗ = 0 σ∗ = 0.0.05 σ∗ = 0.1 σ∗ = 0.15 σ∗ = 0.2

−0.05 0.6174885 0.609253038 0.586111851 0.552019438 0.511710031

−0.025 0.559934754 0.55306281 0.533660493 0.504831619 0.470378391

0 0.511437589 0.50562678 0.489153677 0.464497747 0.434756641

0.025 0.469992119 0.46502534 0.450895313 0.42961117 0.403726307

0.05 0.434159568 0.429876144 0.417652383 0.399135514 0.376450947


FIGURE 4.7: DL LSC versus SD σ∗for different values of skewness ε∗


ε∗ σ∗ = 0 σ∗ = 0.0.05 σ∗ = 0.1 σ∗ = 0.15 σ∗ = 0.2

−0.05 0.574532021 0.56658864 0.544373185 0.511906498 0.473877705

−0.025 0.540313265 0.53358434 0.514621592 0.486537348 0.453103647

0 0.511437589 0.50562678 0.489153677 0.464497747 0.434756641

0.025 0.486612575 0.481518505 0.467008046 0.445102265 0.418388902

0.05 0.464952263 0.460432123 0.447505722 0.427851468 0.40366242


FIGURE 4.8: DL LSC versus mean α∗ for different values of skewness ε∗

TABLE 4.7: Variation in DL LSC with respect to mean α∗ for different values ofskewness ε∗

ε∗ α∗ =−0.05 α∗ =−0.025 α∗ = 0 α∗ = 0.025 α∗ = 0.05

−0.05 0.676616593 0.60359288 0.544373185 0.495255145 0.45379332

−0.025 0.626664165 0.565509556 0.514621592 0.471561585 0.43462839

0 0.586111851 0.533660493 0.489153677 0.450895313 0.417652383

0.025 0.552273371 0.506473272 0.467008046 0.432647676 0.402468797

0.05 0.52344332 0.482888921 0.447505722 0.416371689 0.388776775

Figs.(4.6 and 4.7) represent the variation of DL LSC with respect to SD. It is found that

SD decreases the load carrying capacity. Also, positively skewed roughness and mean

(+ Ve) enhance the adverse effect of SD. Fig.(4.8) indicates that the mean (-ve) registers

a considerable positive effect especially, when the negatively skewed roughness occurs.


FIGURE 4.9: DL LSC versus eccentricity ratio ε for different values of mean α∗

TABLE 4.8: Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of mean α∗

α∗ ε = 0 ε = 0.2 ε = 0.4 ε = 0.6

−0.05 0.05 0.245243461 0.586111851 1.56906215

−0.025 0.05 0.229483014 0.533660493 1.360761214

0 0.05 0.215661637 0.489153677 1.19680674

0.025 0.05 0.203444046 0.450895313 1.064108181

0.05 0.05 0.192569316 0.417652383 0.954405774


FIGURE 4.10: DL LSC versus eccentricity ratio ε for different values of SD σ∗

TABLE 4.9: Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of SD σ∗

σ∗ ε = 0 ε = 0.2 ε = 0.4 ε = 0.6

0 0.05 0.221310216 0.511437589 1.31547764

0.05 0.05 0.219860525 0.50562678 1.283052442

0.1 0.05 0.215661637 0.489153677 1.19680674

0.15 0.05 0.209125466 0.464497747 1.081004072

0.2 0.05 0.200827693 0.434756641 0.958115758


FIGURE 4.11: DL LSC versus eccentricity ratio ε for different values of skewness ε∗

TABLE 4.10: Variation in DL LSC with respect to eccentricity ratio ε for differentvalues of skewness ε∗

ε∗ ε = 0 ε = 0.2 ε = 0.4 ε = 0.6

−0.05 0.05 0.226731133 0.546528366 1.867724777

−0.025 0.05 0.221276414 0.516493439 1.429711799

0 0.05 0.216181741 0.490804451 1.202697188

0.025 0.05 0.211410081 0.468481604 1.057284934

0.05 0.05 0.206929555 0.448834114 0.953466012

The effect of eccentricity ratio presented in Figs(4.9 to 4.11). It is observed that that the

LSC increases due to eccentricity ratio. The initial effect due to eccentricity ratio on

LSC is nominal.


4.4 Conclusion

It is established that for an overall improved performance of the bearing system,

selection of the eccentricity ratio is carefully required. The adverse effect of the

roughness can be minimized by the suitable magnetic strength and eccentricity ratio

while deigning the bearing system.

CHAPTER 5

A Magnetic Fluid Based a Longitudinal

Rough Exponential Slider Bearing

57

A Magnetic Fluid Based a Longitudinal Rough Exponential Slider Bearing 58

5.1 Introduction

The most common and generally encountered hydrodynamic bearing is the SB. In the

frictional devices like clutch plates, automobile transmissions etc use the SB. Many

researchers like [Lord Rayleigh [27], Pinkus and Sternlicht [46], Cameron [10]] have

studied SB for different film shapes. It is established that roughness is developed in

bearing after some run. With the implementation of stochastic approach, Tzeng and

Saibel [57] evaluated the effect of roughness in SB. Christensen and Tonder [13] [11]

[14] analyzed three different models of hydrodynamic lubrication of an SB with

surfaces roughness. The first model is related with one-dimensional LR, the second is

associated to one-dimensional TR and the third deals with the case of uniform,

isotropic roughness. This stochastic approach of Christensen and Tonder [13] [11] [14]

for evaluate the effect of surface roughness was used by many authors [ Guha [18],

Prakash and Tiwari [47], Gupta and Deheri [19] and Patir and cheng[44] ]. A crucial

importance has been allotted to fluid with strong magnetic properties for the last

decade. The performance of a magnetic fluid based inclined SB was studied by

Agrawal [2] and found that the magnetic fluid modified the performance of SB system.

Bhat and Deheri [6] analyzed the performance of an exponential SB with a ferrofluid

lubricant concluded that the magnetic fluid lubrication caused increased LSC slightly

altering the friction on the slider. The analysis of Shah et al. [52] observed the positive

effect of magnetic fluid lubrication over the conventional fluids. Andharia et al. [3] and

Deheri et al. [15] suggested that in case of LR by suitably choosing of the

magnetization parameter, the performance of the bearing could be improved. The

performance of a rough hyperbolic SB under the presence of a ferrofluid lubricant was

studied by Shukla and Deheri [55]. It is found that the LSC in addition, to friction

increase with increasing magnetization. The negatively skewed roughness induced to

increase LSC goes a long way in mitigating the adverse effect of the SD taking

recourse to suitable magnetic strength. Patel and Deheri [38] analyzed the comparison

of several porous structures on the performance of a ferro fluid based short bearing

with TR. It was established that the effect of magnetization is positive, while the

bearing suffered because of the TR. The influence of roughness parameters on the PD

Analysis 59

and LSC in a finite plane SB with LR was analyzed Deheri et al. [36]

Here, it has been discussed that the effect of magnetic fluid through a series of

flowfactors on the behavior of a longitudinal rough exponential SB.

5.2 Analysis

FIGURE 5.1: Exponential Slider Bearing

The bearing configuration is presented in Fig. (5.1). Assuming the slider moves with

the uniform velocity U in the X direction. The length of the bearing is l and breath of

bearing is b with l << b while h1 and h0 are maximum and minimum film thickness

respectively. The film thickness h is taken as


h = h0.e

(l− x

l

)(5.1)

Following the investigations of Agrawal [25] the magnitude M of the magnetic field H

is considered to be

M2 = Kx(l− x) (5.2)

Making use of a stochastic averaging method of Christensen and Tonder [13] [11] [14]

and Agrawal [2] , Bhat [4], Patir [44], the form of modified Reynolds equation is

ddx

[ϕx

h3

12η

ddx

(p−0.5µµ0M2)]= U

2dhdx

(5.3)

The stochastic lubricant film thickness is considered as suggested by Christensen and

Tonder [13] [11] [14] in the form:

H = h(x)+δ

where, h(x) is the lubricant mean film thickness and δ is the deviation from h(x). The

mean α , SD σ and skewness ε associated with random variable δ are governed by the

relations

α = E(δ )

Analysis 61

σ2 = E[(δ −α)2

]

and

ε = E[(δ −α)3

]

where, E denotes the expected value defined by

E () =∫ c−c () f (δ )dδ

Following the averaging process discussed by Andharia et al. [15], Eq.(5.3) reduces to

ddx

[ϕx

112η

1A(h)

ddx

(p−0.5µµ0M2)]= U

2ddx

[1

B(h)

](5.4)

where,

A(h) =1h3

[1− 3α

h−20

α3 + ε +3σ2α

h3 +6α2 +σ2

h2

]

B(h) =1h

[1− α

h− 3σ2α +α3 + ε

h3 +α2 +σ2

h2

]

Making use of equations (5.1) and (5.2) and dimensionless quantities:


X =xl, h∗ =

hh0

, σ∗ =σ

h0, α∗ =

α

h0, ε∗ =

ε

h0, A(h∗) = h3

0.A(h), B(h∗) = h0.B(h),

P =ph2

0Uη l

, µ∗ =µ0µlh2

02ηU

,

equation (4.4) transforms to,

ddX

[ϕx

1A(h∗)

ddX

(P−µ∗X(1−X))

]= 6

ddX

[1

B(h∗)

](5.5)

An experimental relation for Φx obtained by Patir [44] is as under,

Φx = 1+C.H−r(γ > 1) (5.6)

And the relation between constants C, r and γ given as below table,

TABLE 5.1: Relation between C, r and H

γ r C H

3 1.5 0.225 H > 0.5

6 1.5 0.520 H > 0.5

9 1.5 0.870 H > 0.5

Analysis 63

Solving Eq.(5.5) under the boundary conditions:

P = 0 at X = 0 and X = 1

one obtains the expression for DL PD as:

P = µ∗(X−X2)+

∫ X

0

A(h∗)Φx

[6

B(h∗)−Q∗

]dX (5.7)

where,

Q∗ =

∫ 10

6A(h∗)ΦxB(h∗)∫ 1

0A(h∗)

Φx

Then, the DL DSC is expressed as:

W =wh2

0µUl

=∫ 1

0P(X)dX (5.8)


5.3 Results and discussion

It is observed from Eq.(5.7) that the PD increases by

µ∗X(1−X)

as compared to conventional lubricant based bearing system. Therefore, the LSC

enhances. The variation of DL LSC with respect to magnetization parameter is

presented in Figs.(5.2 to 5.5) for various values of SD σ∗, skewness ε∗, mean α∗ and

RPP γ respectively. It is seen the effect of magnetization is positive on LSC. It is

noticed that for smaller values of SD σ∗ the effect of magnetization parameter on the

variation of DL LSC is marginal.(Fig.(5.2)) It is observed that skewness (+ve)

decreases the LSC while (-ve) skewness increases the LSC.(Fig.(5.3)) The mean

follows the same trends of skewness.(Fig.(5.4))

The variation of the DL LSC versus the SD σ∗ for several values of the skewness ε∗

and mean α∗ is presented in Figs.(5.6) and (5.7). The combined effect of mean and

skewness is illustrated in Fig.(5.8). It is clear from these graphs that increased the LSC

due to SD σ∗, gets further increased due to the mean (-ve) and negatively skewed

roughness.

Results and discussion 65

FIGURE 5.2: DL LSC versus magnetic parameter µ∗ for different values of SD σ∗


σ∗ µ∗ = 0.02 µ∗ = 0.04 µ∗ = 0.06 µ∗ = 0.08 µ∗ = 0.1

0.0 0.595551501 0.627551501 0.659551501 0.691551501 0.723551501

0.025 0.597354466 0.629354466 0.661354466 0.693354466 0.725354466

0.05 0.602744288 0.634744288 0.666744288 0.698744288 0.730744288

0.075 0.611664802 0.643664802 0.675664802 0.707664802 0.739664802

0.1 0.624025824 0.656025824 0.688025824 0.720025824 0.752025824


FIGURE 5.3: DL LSC versus magnetic parameter µ∗ for different values of skewnessε∗


ε∗ µ∗ = 0.02 µ∗ = 0.04 µ∗ = 0.06 µ∗ = 0.08 µ∗ = 0.1

−0.025 0.624025824 0.656025824 0.688025824 0.720025824 0.752025824

−0.01 0.562081181 0.594081181 0.626081181 0.658081181 0.690081181

0.0 0.515569884 0.547569884 0.579569884 0.611569884 0.643569884

0.01 0.463410529 0.495410529 0.527410529 0.559410529 0.591410529

0.025 0.370108459 0.402108459 0.434108459 0.466108459 0.498108459


FIGURE 5.4: DL LSC versus magnetic parameter µ∗ for different values of mean α∗


α∗ µ∗ = 0.02 µ∗ = 0.04 µ∗ = 0.06 µ∗ = 0.08 µ∗ = 0.1

−0.05 0.624025824 0.656025824 0.688025824 0.720025824 0.752025824

−0.025 0.573123924 0.605123924 0.637123924 0.669123924 0.701123924

0.0 0.523451626 0.555451626 0.587451626 0.619451626 0.651451626

0.025 0.474822774 0.506822774 0.538822774 0.570822774 0.602822774

0.05 0.427008756 0.459008756 0.491008756 0.523008756 0.555008756


FIGURE 5.5: DL LSC versus magnetic parameter µ∗ for different values of RPP γ

TABLE 5.5: Variation in DL LSC with respect to magnetic parameter µ∗ for differentvalues of RPP γ

γ µ∗ = 0.02 µ∗ = 0.04 µ∗ = 0.06 µ∗ = 0.08 µ∗ = 0.1

3 0.642135275 0.674135275 0.706135275 0.738135275 0.770135275

6 0.624025824 0.656025824 0.688025824 0.720025824 0.752025824

9 0.604024703 0.636024703 0.668024703 0.700024703 0.732024703


FIGURE 5.6: DL LSC versus SD σ∗ for different values of skewness ε∗


ε∗ σ∗ = 0 σ∗ = 0.025 σ∗ = 0.05 σ∗ = 0.075 σ∗ = 0.1

−0.025 0.723551501 0.725354466 0.730744288 0.739664802 0.752025824

−0.01 0.658935373 0.660914079 0.666823816 0.676587241 0.690081181

0 0.609989005 0.612129082 0.618515253 0.629047996 0.643569884

0.01 0.554588546 0.556944984 0.563968639 0.575526651 0.591410529

0.025 0.45391985 0.45677389 0.465258833 0.47915182 0.498108459


FIGURE 5.7: DL LSC versus SD σ∗ for different values of mean α∗

TABLE 5.7: Variation in DL LSC with respect to SD σ∗ for different values of meanα∗

α∗ σ∗ = 0 σ∗ = 0.025 σ∗ = 0.05 σ∗ = 0.075 σ∗ = 0.1

−0.05 0.723551501 0.725354466 0.730744288 0.739664802 0.752025824

−0.025 0.674734813 0.676402329 0.681390051 0.689654184 0.701123924

0 0.627355129 0.628874678 0.633422357 0.640965653 0.651451626

0.025 0.581198451 0.582559533 0.586635053 0.593402029 0.602822774

0.05 0.536011309 0.537205195 0.540781585 0.546724752 0.555008756


FIGURE 5.8: DL LSC versus skewness ε∗ for different values of mean α∗

TABLE 5.8: Variation in DL LSC with respect to skewness ε∗ for different values ofmean α∗

α∗ ε∗ = 0 ε∗ = 0.025 ε∗ = 0.05 ε∗ = 0.075 ε∗ = 0.1

−0.025 0.752025824 0.690081181 0.643569884 0.591410529 0.498108459

−0.01 0.701123924 0.636938285 0.588541529 0.534014877 0.435664379

0 0.651451626 0.58519061 0.535055807 0.478348856 0.37532719

0.01 0.602822774 0.534635515 0.482899439 0.424190796 0.316876398

0.025 0.555008756 0.485013715 0.431788293 0.371227796 0.259949673


5.4 Conclusion

The investigation shows that the magnetization may go a long way in overcoming the

adverse effect of roughness. However, the study strongly shows that the roughness

aspect must be duly dealt with while designing the bearing system, even though a

suitable magnetic strength is applied.

CHAPTER 6

Several PDF Related With the

Roughness Characteristics on the

Performance of Longitudinal Rough

Slider Bearing

73

Several PDF Related With the Roughness Characteristics on the Performance ofLongitudinal Rough Slider Bearing 74

6.1 Introduction

The efficiency of the bearings is affected by various aspects like viscosity, roughness,

density etc. It is clear that mostly it was assumed that smooth bearing surfaces were

utilized in studies. It is essential for us to examine the roughness for some basic problem

like friction, LSC, PD. Because of its important role, roughness is the area of interest

for many scholars in studies. Usually picks and valleys of different size scattered over

the bearing surfaces are found. The scattered picks and valleys show the necessity

of the stochastical approach for evaluation of surface roughness. According to Bharat

Bhushan [8] the evaluation of surface roughness can be done by using the probability

distribution. With the implementation of stochastic approach, Tzeng and Saibel [57]

evaluated the effect of surface roughness in SB by using a beta PDF and it became more

realistic representation of bearing. The approach of Tzeng and Saibel [57] is applied

and redefined by Christensen and Tonder [13] [11] [14] and introduced the stochastic

model of surface roughness depends on general PDF. Surface roughness evaluation has

been studied by application of stochastic model of Christensen and Tonder [13] [11]

by many scholars such as Sinha and Adamu [56], Patel and Deheri [37], Prakash and

Tiwari [48], K. Gururajan and J. Prakash [20] and Deheri et al. [36]. In all the above

studied, Pseusdo normal PDF is used to evaluate the roughness characteristics.

So, here it is going to be discussed the effect of several PDF related with the roughness

characteristics on the performance of longitudinal rough SB.

6.2 Analysis

The generalized Reynolds equations for a rough bearing system is

∂

∂ z

(H3 ∂ p

∂ z

)+

∂

∂x

(H3 ∂ p

∂x

)= 6Uη

∂H∂x

(6.1)

Analysis 75

The stochastic lubricant film thickness is taken as suggested by Christensen and Tonder

[13] [11] [14] in the form:

H = h(x)+δ (6.2)

where, h(x) is the lubricant mean film thickness and δ is the deviation from h(x).

Applying the expectancy operator E, on both sides of Eq.(6.1) following Christensen

[13] [11] [14], the Eq.(6.1) reduce to

∂

∂x

(E(H3)

∂ p∂x

)+

∂

∂ z

(E(H3)

∂ p∂ z

)= 6ηU

∂E(H)

∂x(6.3)

Here E denotes the expected value given by

E() =∫ c

−c() f (δ )dδ (6.4)

where, f is the PDF of random variable δ . In order to achieve comparitive study of

different kinds of PDF associated roughness, the PDF are considered a given below:

1 Pseudo-normal distribution:

f (δ ) =

35

32c7 (c2−δ 2)3 if − c≤ δ ≤ c,

0 if elsewhere.


2 Beta distribution:

f (δ ) =

15

16c5

(c2−δ 2)2 if − c≤ δ ≤ c,

0 if elsewhere.

3 Rectangular distribution:

f (δ ) =

12c if − c≤ δ ≤ c,

0 if elsewhere.

FIGURE 6.1: Several Distribution Function

Analysis 77


The bearing geometry is given in Fig.(6.2). Here we assume that SB is infinitely long in

z direction. So there is no variation in pressure in z direction. Therefore, the∂ p∂ z

term

in Eq.(6.3) can be neglected.

The Reynolds equation for PD in this case will be

ddx

(E(H3)

d pdx

)= 6ηU

dE(H)

dx(6.5)

The associated boundary conditions are,

p = 0 at x = 0 and x = l

E(H3) and E(H) are given below for several PDF.

1 Pseudo-normal distribution:

E(H3) =∫ c

−c(h+δ )3 f (δ )dδ = h3 +

2512

hc2

E(H) =∫ c

−c(h+δ ) f (δ )dδ = h


2 Beta distribution:

E(H3) =∫ c

−c(h+δ )3 f (δ )dδ = h3 +

37

hc2

E(H) =∫ c

−c(h+δ ) f (δ )dδ = h

3 Rectangular distribution:

E(H3) =∫ c

−c(h+δ )3 f (δ )dδ = h3 +hc2

E(H) =∫ c

−c(h+δ ) f (δ )dδ = h

Finally, the form of modified Reynolds equation using the above expression for E(H)

and E(H3) is

ddx

((h3 +αhc2) d p

dx

)= 6ηU

dhdx

(6.6)

where, α is 1 for Rectangular distribution,2512

for Pseudo-normal distribution and37

for

Beta distribution.

Analysis 79

Introducing the DL quantities

X =xl,h∗ =

hh0

,c∗ =ch0

,P =ph2

06Uη l

the Eq.(6.6) reduces to

ddX

((h∗3 +αh∗c∗2

) dPdX

)=

dh∗

dX(6.7)

In view of the boundary conditions

P = 0 at X = 0 and X = 1


P =1

(τ−1)

[h∗m

αc∗2log

h∗√h∗2 +αc∗2

− 1√αc∗

tan−1 h∗√αc∗

]+C1 (6.8)

where,

τ =h1

h0


C1 =1

(τ−1)

[(1√αc∗

tan−1 τ√αc∗

)−(

h∗mαc∗2

logτ√

τ2 +αc∗2

)]

h∗m =√

αc∗tan−1

( √αc∗(τ−1)τ +αc∗2

)

log

τ

√αc∗2 +1αc∗2 + τ2

The DL LSC is obtained as:

W =1

(τ−1)2

(h∗m

αc∗2

(τ(logτ−1)+2− 1

2

(log

(τ2 +αc∗2)τ

1+αc∗2

- 2(τ−1)+2√

αc∗tan−1√

αc∗(τ−1)τ +αc∗2

))−(

τ√αc∗

tan−1 τ√αc∗

-1√αc∗

tan−1 1√αc∗

+ log

√1

αc∗2+1− log

√τ2

αc∗2+1))

+C1 (6.9)

The DL friction force is obtained by

F =∫ 1

0

(1h∗

+h∗

2dPdx

)dX (6.10)

Lastly, the coefficient of friction is obtained from the relation:

Results and Discussion 81

µ =FW

(6.11)

From the experimental study of N.S.Patel [42], J. Kishigami [34], dimensional

parameters like surface roughness height c, maximum film thickness h1 and minimum

film thickness h0 are selected which is given in below table.

TABLE 6.1: Selection of DL parameters

Dimensional parameters

c 0.3 µm to 0.8 µm

h0 0.06 mm to 0.15 mm

h1 0.12 mm to 0.37 mm

DL parameters

c∗ =ch0

0.008 to 0.013

τ =h1

h02 to 2.5

6.3 Results and Discussion

The effect of roughness on PD and friction is found significantly. The bearing surfaces

come in more contact when the PD is not in proper order and the deterioration is

caused. The analysis of PD and friction with reference to different roughness

parameters.


FIGURE 6.3: DL PD versus RPP α for different values of surface roughness height c∗

TABLE 6.2: Variation in DL PD with respect to RPP α for different values of surfaceroughness height c∗

c∗ α = 37 α = 1 α = 25

12

0.008 239.4870151 156.6236581 108.3720338

0.009 212.8266192 139.1703443 96.28002911

0.01 191.4982862 125.2076865 86.60642251

0.011 174.0478155 113.7836869 78.69165061

0.012 159.50574 104.2636803 72.09600446

0.013 147.2008903 96.20828324 66.5150702


FIGURE 6.4: DL PD versus RPP α for different values of FTR τ

TABLE 6.3: Variation in DL PD with respect to RPP α for different values of FTR τ

τ α = 37 α = 1 α = 25

12

2 239.3131262 156.4498612 108.1982772

2.1 217.5801807 142.2499453 98.3848703

2.2 199.4680642 130.4153536 90.20570288

2.3 184.1412968 120.400338 83.28373871

2.4 171.0031002 111.8150714 77.34965917

2.5 159.6158303 104.3736744 72.2059576

The effect of different kinds of roughness patterns associated with PDF on PD is

presented in Fig.(6.3 and 6.4). It seems that the PD is more found in Beta distribution

compare to other two distributions.


FIGURE 6.5: DL PD versus surface roughness height c∗ for different values of RPP α

TABLE 6.4: Variation in DL PD with respect to surface roughness height c∗ fordifferent values of RPP α

α c∗ = 0.008 c∗ = 0.009 c∗ = 0.01 c∗ = 0.011 c∗ = .012 c∗ = 0.013

37 239.4870 212.8266 191.4982 174.0478 159.505 147.2008

1 156.6236 139.1703 125.2076 113.7836 104.2636 96.20822512 108.3720 96.2800 86.6064 78.6916 72.0960 66.5150


FIGURE 6.6: DL PD versus FTR τ for different values of RPP α

TABLE 6.5: Variation in DL PD with respect to FTR τ for different values of RPP α

α τ = 2 τ = 2.1 τ = 2.2 τ = 2.3 τ = 2.4 τ = 2.5

37 239.3131 217.5801 199.4680 184.1412 171.0031 159.6158

1 156.4498 142.2499 130.4153 120.4003 111.8150 104.37362512 108.1982 98.3848 90.20570 83.2837 77.3496 72.2059

The observation from Fig.(6.5) is that there is decrease in PD for all roughness patterns

associated with PDF when deviation from the mean film thickness gets increased. In

Fig.(6.6), the effect of FTR on PD is presented. it clear that the PD reduces when the

FTR gets increased in all roughness patterns associated with PDF.


FIGURE 6.7: Coefficient of friction µ versus RPP α for surface roughness height c∗

TABLE 6.6: Variation in coefficient of friction µ with respect to RPP α for differentvalues of surface roughness height c∗

c∗ α = 37 α = 1 α = 25

12

0.008 0.016567764 0.025192686 0.036149257

0.009 0.018618672 0.028294925 0.040570231

0.01 0.020665118 0.031386742 0.044969461

0.011 0.022707101 0.034468132 0.049346937

0.012 0.02474462 0.037539093 0.053702651

0.013 0.026777674 0.04059962 0.05803659


FIGURE 6.8: Coefficient of friction µ versus RPP α for FTR τ

TABLE 6.7: Variation in coefficient of friction µ with respect to RPP α for differentvalues of FTR τ

τ α = 37 α = 1 α = 25

12

2 0.015260249 0.023177586 0.033207623

2.1 0.017346365 0.026346457 0.037748609

2.2 0.019533591 0.029668386 0.042507933

2.3 0.02182192 0.033143356 0.04748556

2.4 0.024211344 0.036771346 0.052681446

2.5 0.026701852 0.040552333 0.058095542


FIGURE 6.9: Coefficient of friction µ versus FTR τ for different values of surfaceroughness height c∗

TABLE 6.8: Variation in DL PD with respect to FTR τ for different values of surfaceroughness height c∗

c∗ τ = 2 τ = 2.1 τ = 2.2 τ = 2.3 τ = 2.4 τ = 2.5

0.008 0.0186036 0.0211469 0.0238133 0.0266028 0.0295153 0.0325509

0.009 0.0208944 0.0237510 0.0267457 0.0298785 0.0331495 0.0365584

0.01 0.0231775 0.0263464 0.0296683 0.0331433 0.0367713 0.0405523

0.011 0.0254530 0.0289331 0.0325811 0.0363970 0.0403808 0.0445325

0.012 0.0277207 0.0315110 0.0354840 0.0396397 0.0439781 0.0484990

0.013 0.0299807 0.0340802 0.0383771 0.0428714 0.0475630 0.0524519

It is observed from Figs.(6.7 to 6.9) that the surface roughness height and FTR have an

adverse effect in the sense that the coefficient of friction increases due to them.

Conclusion 89

6.4 Conclusion

In the present theoretical investigation, it is found that the FTR and RPP can have

crucial role in improving the bearing performance. So the performance characteristics

can be made better by selecting design parameter carefully. Moreover the bearing

performance made better in Beta distribution compere to other distributions.

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[26] J R Lin, R F Lu and W H Liao, Analysis of magneto-hydrodynamic squeeze film characteristicsbetween curved annular plates, Industrial Lubrication and Tribology, 56(5), (2004), 300–305.

[27] R Lord, I. Notes on the theory of lubrication, The London, Edinburgh, and Dublin PhilosophicalMagazine and Journal of Science, 35, (1918), 1–12.

[28] R F Lu, R D Chien and J R Lin, Effects of fluid inertia in magneto-hydrodynamic annular squeezefilms, Tribology International, 39(3), (2006), 221–226.

[29] G Nada and T Osman, Static performance of finite hydrodynamic journal bearings lubricated bymagnetic fluids with couple stresses, Tribology letters, 27(3), (2007), 261–268.

[30] N Naduvinamani, P Hiremath and G Gurubasavaraj, Squeeze film lubrication of a short porousjournal bearing with couple stress fluids, Tribology International, 34(11), (2001), 739–747.

[31] N Naduvinamani and B Kashinath, Surface roughness effects on the static and dynamic behaviourof squeeze film lubrication of short journal bearings with micropolar fluids, Proceedings of theInstitution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 222(2), (2008),121–131.

[32] J L Neuringer and R E Rosensweig, Ferrohydrodynamics, The Physics of Fluids, 7(12), (1964),1927–1937.

[33] S Odenbach, Recent progress in magnetic fluid research, Journal of physics: condensed matter,16(32), (2004), R1135.

[34] T Ohkubo and J Kishigami, Accurate measurement of gas-lubricated slider bearing separation usingvisible laser interferometry, Journal of tribology, 110(1), (1988), 148–155.

[35] T Osman, G Nada and Z Safar, Effect of using current-carrying-wire models in the design ofhydrodynamic journal bearings lubricated with ferrofluid, Tribology Letters, 11(1), (2001), 61–70.

[36] G C Panchal, H C Patel and G Deheri, INFLUENCE OF SURFACE ROUGHNESS THROUGH ASERIES OF FLOW FACTORS ON THE PERFORMANCE OF A LONGITUDINALLY ROUGHFINITE SLIDER BEARING, Annals of the Faculty of Engineering Hunedoara, 14(2), (2016), 227.

[37] C Patel and G Deheri, Characteristics of lubrication at nano scale on the performance of transverselyrough slider bearing, Mechanics, 80(6), (2009), 64–71.

[38] J R Patel and G Deheri, A comparison of porous structures on the performance of a magnetic fluidbased rough short bearing, Tribology in industry, 35(3), (2013), 177–189.

[39] N Patel, D Vakharia and G Deheri, A study on the performance of a magnetic-fluid-basedhydrodynamic short journal bearing, ISRN mechanical engineering, 2012.

[40] N Patel, D Vakharia and G Deheri, Magnetic fluid lubrication of finite journal bearing; 3-D analysisusing FDM, British Journal of Applied Science & Technology, 4(205), (2014), 177.

References 92

[41] N Patel, D Vakharia, G Deheri and H Patel, The performance analysis of a magnetic fluid-basedhydrodynamic long journal bearing, in: Proceedings of International Conference on Advances inTribology and Engineering Systems, Springer (2014), 117–126.

[42] N S Patel, D Vakharia and G Deheri, Hydrodynamic journal bearing lubricated with a ferrofluid,Industrial Lubrication and Tribology, 69(5), (2017), 754–760.

[43] S Patel, G Deheri and J Patel, Ferrofluid Lubrication of a Rough Porous Hyperbolic Slider Bearingwith Slip Velocity., Tribology in Industry, 36(3).

[44] N Patir and H Cheng, An average flow model for determining effects of three-dimensionalroughness on partial hydrodynamic lubrication, Journal of lubrication Technology, 100(1), (1978),12–17.

[45] N Patir and H Cheng, Application of average flow model to lubrication between rough slidingsurfaces, Journal of Lubrication Technology, 101(2), (1979), 220–229.

[46] O Pinkus and B Sternlicht, Theory of hydrodynamic lubrication.

[47] J Prakash and K Tiwari, Lubrication of a porous bearing with surface corrugations, Journal oflubrication technology, 104(1), (1982), 127–134.

[48] J Prakash and K Tiwari, Roughness effects in porous circular squeeze-plates with arbitrary wallthickness, Journal of Lubrication Technology, 105(1), (1983), 90–95.

[49] A Raj and P Sinha, Transverse roughness in short journal bearing under dynamic loading, Tribologyinternational, 16(5), (1983), 245–251.

[50] O Reynolds, IV. On the theory of lubrication and its application to Mr. Beauchamp towerexperiments, including an experimental determination of the viscosity of olive oil, PhilosophicalTransactions of the Royal Society of London, 177, (1886), 157–234.

[51] R C Shah and M Bhat, Ferrofluid squeeze film in a long journal bearing, Tribology International,37(6), (2004), 441–446.

[52] R C Shah, S Tripathi and M Bhat, Magnetic fluid based squeeze film between porous annular curvedplates with the effect of rotational inertia, Pramana, 58(3), (2002), 545–550.

[53] M Shimpi and G Deheri, Effect of deformation in magnetic fluid based transversely rough shortbearing, Tribology-Materials, Surfaces & Interfaces, 6(1), (2012), 20–24.

[54] M Shimpi and G Deheri, Effect of Bearing Deformation on the Performance of a Magnetic FluidBased Infinitely Rough Short Porous Journal Bearing, in: Proceedings of International Conferenceon Advances in Tribology and Engineering Systems, Springer (2014), 19–33.

[55] S Shukla and G Deheri, Surface Roughness Effect on the Performance of a Magnetic Fluid BasedHyperbolic Slider Bearing, International Journal of Engineering Research and Applications, 1(3),(2011), 948–962.

[56] P Sinha and G Adamu, THD analysis for slider bearing with roughness: special reference to loadgeneration in parallel sliders, Acta mechanica, 207(1-2), (2009), 11–27.

[57] S Tzeng and E Saibel, Surface roughness effect on slider bearing lubrication, Asle Transactions,10(3), (1967), 334–348.

[58] H Urreta, Z Leicht, A Sanchez, A Agirre, P Kuzhir and G Magnac, Hydrodynamicbearing lubricated with magnetic fluids, 11th Conference on Electrorheological Fluids andMagnetorheological Suspensions.

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[59] H Urreta, Z Leicht, A Sanchez, A Agirre, P Kuzhir and G Magnac, Hydrodynamic bearinglubricated with magnetic fluids, Journal of intelligent material systems and structures, 21(15),(2010), 1491–1499.

[60] P Verma, Magnetic fluid-based squeeze film, International Journal of Engineering Science, 24(3),(1986), 395–401.

List of Publications

List of Publications Arising From the Thesis

1. Magnetic Fluid Lubrication of an Infinitely Long Slider Bearing with Rough Surfaces, Journal of

the Serbian Society for Computational Mechanics,(UDC: 621.822.5:532.54), Vol. 9, No. 2, 2015,

pp. 10-18.

2. Characteristics of magnetic fluid lubrication on the perfomance of a transversely rough

hydrodynamic short journal bearing, International Journal of Mechanical Engineering and

Technology, ISSN Print: 0976-6340 , ISSN Online: 0976-6359, Volume 8, Issue 5, May 2017,

pp. 1110 to 1118.

3. Effect of magnetic fluid through a series of flowfactors on the behavior of a longitudinally rough

exponential slider bearing International Journal of Current Engineering and Scintific Reasearch

, ISSN (PRINT): 2393-8374, (ONLINE): 2394-0697, Volume-5, Issue-3, 2018.

Details of the Work Presented in Conference From the Thesis

1. The Theoretical Investigation of Several Probability Density Function Associated with the

Roughness Characteristics for the Performance of Rough Slider Bearing, Lecture Notes in

Mechanical Engineering (Springer): Proceedings of the 1st International Conference on

Numerical Modelling in Engineering-2018, ISSN-2195-4356.

94

The Theoretical Investigation of SeveralProbability Density Function Associated

with the Roughness Characteristicsfor the Performance of Rough

Slider Bearing

Himanshu C. Patel1(B), Mehul P. Patel2(B), Nimeshchandra S. Patel3(B),and G. M. Deheri4(B)

1 L. D. College of Engineering, Ahmedabad 380009, Gujarat, [email protected]

2 K. D. Polytechnic, Patan 384265, Gujarat, [email protected]

3 Faculty of Technology, D. D. University, Nadiad 387001, Gujarat, [email protected]

4 S. P. University, Vallabh Vidyanagar 388120, Gujarat, [email protected]

Abstract. The paper examines the effects of surface roughness on filmpressure in hydrodynamic slider bearings. The estimation of the effectof surface roughness has been done on the basis of the stochastic theory,which is developed by Christensen and Tonder model. Three differenttypes of probability density functions have been used to evaluate theeffect of the longitudinal surface roughness leading to a comparison ofthe bearing performances. After solving the Reynolds type equation thepressure distribution is obtained. This is then used to compute the loadcarrying capacity. Besides, friction has also been obtained using differentmodels. This study studies various mathematical models for estimationof pressure distribution, which may be helpful to the engineers for betterdesign of the bearing systems. . . .

Keywords: Reynolds’ equation · Longitudinal roughnessProbability density functions · Fluid film pressureLoad carrying capacity

1 List of Symbols (SI Units)

h Nominal film thickness (m)h∗ Nominal film thickness (Dimensionless)hs Deviation from the mean film thickness (m)h0 Minimum film thickness at the trailing edge of slider bearing (m)h1 Maximum film thickness at the trailing edge of slider bearing (m)H Film thickness (m)c⃝ Springer Nature Singapore Pte Ltd. 2019M. Abdel Wahab (Ed.): NME 2018, LNME, pp. 179–190, 2019.https://doi.org/10.1007/978-981-13-2273-0_15

Publication 1 95

180 H. C. Patel et al.

τ Film thickness ratio (Dimensionless)l Length of slider bearing (m)c Maximum deviation from the mean film thickness (m)c∗ Maximum deviation from the mean film thickness (Dimensionless)

f(hs) Probability density function of combined roughness amplitude-hs (m−1)η Viscosity of lubricant (Kgm−1s−1)U Velocity of bearing surface in X-Direction

m

sE() Expectancy operator

E(p) Expected value of the mean pressure levelN

m2

E(p∗) Expected value of the mean pressure level (Dimensionless)E(W ∗) Expected value of Load carrying capacity (Dimensionless)E(F ∗) Expected value of Friction (Dimensionless)

µ Friction coefficient (Dimensionless)

2 Introduction

Slider bearings are used in various types of machines due to their stability, dura-bility and high load carrying capacity. Different characteristics of such bearinghave been investigated by many studies. The factors like surface roughness, vis-cosity and density of fluid affect the performance of the bearings.

It has been observed that smooth bearing surfaces are considered in mostof the theoretical investigations. Evaluation of surface roughness is significantfor many fundamental problems such as friction, load carrying capacity, contactdeformation, heat and electric current conditions, tightness of contact joints andpositional accuracy. Surface roughness has been the subject of experimental andtheoretical investigations for many decades due to its crucial role. Generally wefind the surface consists of picks and valleys of different lateral and vertical sizes,which is distributed randomly over the surface. The randomness suggests thatstatistical methods of roughness characterization must be resorted. The proba-bility distribution is one of the characteristics of a rough surface [1]. Tzeng andSaibel [2] introduced stochastic concept and succeeded in carrying out an anal-ysis of the effect of roughness of surfaces of a slider bearing on the load carryingcapacity and friction force by using a beta probability density function for therandom variable characterizing the roughness. This distribution is approximatesthe Gaussian distribution to a good degree of accuracy for certain particularcases. Christensen and Tonder [3–5] developed and modified the approach ofTzeng and Saibel [2] proposed a comprehensive general analysis for the surfaceroughness based on a general probability density function.

Several researchers such as (Prakash and Tiwari [6], Patel and Deheri [7]Sinha and Adamu [8] Gururajan and Prakash [9] Deheri et al. [10]) have inves-tigated the effect of surface roughness on the bearing performance by using theapproach of Christensen and Tonder [3–5]. In all these analyses, the probabilitydensity function for the random variable characterizing the surface roughnesswas considered as Pseudo-normal function.

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Tribology 181

There are very few studies where several probability density functions wereused for the analysis of effect of roughness on the performance of bearing system.So, here it has been proposed to discuss the effect of several probability densityfunction associated with the roughness characteristics on the performance oflongitudinal rough slider bearing.

3 Analysis

The generalized Reynolds equations for a bearing system under the usualassumptions of hydrodynamic lubrication is

∂

∂x

(H3 ∂p

∂x

)+

∂

∂z

(H3 ∂p

∂z

)= 6ηU

∂H

∂x(1)

The film geometry of the fluid film is divided into two parts. The first partis nominal film thickness, which is constant. Owing to surface roughness, thesecond part of film geometry comes into existence, which is measured from thenominal mean level and it is considered as a randomly varying quantity.

In this model, roughness is assumed to have the form of long, narrow peaksand valleys running in the direction of sliding.

Thus, following Christensen and Tonder [3–5] one can describe the film thick-ness as

H = h+ hs (2)

The film thickness component hs is the deviation from the mean film thickness.The details can be seen from Christensen [3–5].

Applying the expectancy operator E, on both sides of Eq. (1) following Chris-tensen [3–5], the Eq. (1) reduce to

∂

∂x

(E(H3)

∂E(p)∂x

)+

∂

∂z

(E(H3)

∂E(p)∂z

)= 6ηU

∂E(H)∂x

(3)

This is the Reynolds equation for rough bearing for longitudinal roughness. Herethe Expectancy operator E is defined by

E() =∫ c

−c()f(hs)dhs (4)

where f is the probability density function of random variable hs. With a viewof getting comparative study of various types of probability density functioninduced roughness, we consider the following probability density distribution:

1. Rectangular distribution:

f(hs) =

{12c if c ≤ hs ≤ c,

0 if elsewhere.(5)

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2. Pseudo-normal distribution

f(hs) =

{35

32c7 (c2 − h2

s)3 if c ≤ hs ≤ c,

0 if elsewhere.(6)

3. Beta distribution:

f(hs) =

⎧⎪⎨

⎪⎩

1516c

(1 − h2

s

c2

)2

if c ≤ hs ≤ c,

0 if elsewhere.(7)

where c is the maximum deviation from the mean film thickness. It is worthremembering that Pseudo-normal distribution has been adopted in most of theinvestigations. It is not easy to deal with the mathematical analysis of Reynoldsequation for variable film thickness geometry using Eq. (3), which is similar tothe usual Reynolds equation for a smooth bearing, although it is possible tocarry mathematical analysis in the limiting case, of infinitely long bearing aswell as short bearing.

The bearing is infinitely long in the Z direction, is presented in Fig. 2. So thepressure variations in the Z direction are nominal compared to sliding direction

and can be neglected. Therefore, the∂

∂zE(p)term in Eq. (3) can be dropped.

Finally, the governing Reynolds equation turns out to be

d

dx

(E(H3)

dE(p)dx

)= 6ηU

dE(H)dx

(8)

The boundary conditions are,

E(p) = 0, at x = 0 and x = l (9)

The evaluation of expected value of various film thickness functions isrequired for the calculation of mean pressure and other bearing characteristics.These are given below.

1. Rectangular distribution:

E(H) =∫ c

−c(h+ hs)f(hs)dhs = h

E(H3) =∫ c

−c(h+ hs)3f(hs)dhs = h3 + hc2

2. Normal distribution

E(H) =∫ c


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Tribology 183

Fig. 1. Several distribution functions

Fig. 2. Infinitely long slider bearing

E(H3) =∫ c

−c(h+ hs)3f(hs)dhs = h3 +

2512

hc2

3. Beta distribution:

E(H) =∫ c


E(H3) =∫ c

−c(h+ hs)3f(hs)dhs = h3 +

37hc2

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Using the expression for E(H) and E(H3) Eq. (8) can be written as

d

dx

((h3 + αhc2

) dE(p)dx

)= 6ηU

dh

dx(10)

where, α is 1 for Rectangular distribution,2512

for Pseudo-normal distribution

and37for Beta distribution.

Introduction of dimensionless quantities

X =x

l, h∗ =

h

h0, c∗ =

c

h0, E(p∗) =

E(p)h20

6Uηl

leads to the dimensionless Reynolds equation:

d

dX

((h∗3

+ αh∗c∗2) dE(p∗)

dX

)=

dh∗

dX(11)


E(p∗) = 0, at X = 0 and X = 1 (12)

Equation (11) when integrated with respect to X yields

dE(p∗)dX

=[

h∗ − (h∗)mh∗3 − αh∗c∗2

](13)

where, subscript m refers to the condition at the point wheredE(p∗)dX

= 0Using the boundary conditions, Eq. (13) can be integrated to give non-

dimensional mean pressure as

E(p∗) =1

(1 − τ)

[1√αc

tan−1 h∗√

αc− h∗

m

αc∗2 logh∗

√h∗2 + αc∗2

]+ C1 (14)

where,

h∗m =

√αc

tan−1

(√αc(τ − 1)αc∗2 + τ

)

log

⎛

⎝τ

√1 + αc∗2

τ2 + αc∗2

⎞

⎠

C1 =1

(1 − τ)

[(h∗m

αc∗2 logτ√

τ2 + αc∗2

)−

(1√αc∗ tan

−1 τ√αc∗

)]

τ =h1

h0The non-dimensional load-carrying capacity is obtained as

E(W∗) =∫ 1

0PdX

=1

(τ − 1)2

⎡

⎣h∗m

αc∗2

⎧⎨

⎩τ(logτ −1)+2 − −1

2

⎧⎨

⎩log(τ2 + αc∗2)τ

1 + αc∗2− 2(τ −1)+2

√αc∗tan−1

( √αc∗(τ − 1)

τ + αc∗2

)⎫⎬

⎭

⎫⎬

⎭

−

⎧⎪⎨

⎪⎩

τ√

αc∗tan−1

(τ

√αc∗

)−

1√

αc∗tan−1

(1

√αc∗

)+ log

⎛

⎝√

1

αc∗2+ 1

⎞

⎠ − log

⎛

⎜⎝

√√√√ τ2

αc∗2+ 1

⎫⎪⎬

⎪⎭

⎤

⎥⎦ + C1

(15)

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Tribology 185

The non-dimensional friction force is determined by

E(F ∗) =∫ 1

0

(h∗

2dP

dx+

1h∗

)dX (16)

Lastly, the coefficient of friction is given by

µ =E(F ∗)E(W ∗)

(17)

Here, the selection of dimensional parameters like minimum film thickness h0,maximum film thickness h1, surface roughness height of slider bearing c arebased on experimental study of Patel [11], Kishigami [12] (Table 1).

Table 1. Selection of non-dimensional parameters

Range of dimensional parameters

h0 0.06 mm to 0.15 mm

h1 0.12 mm to 0.37 mm

c 0.3 µm to 0.8 µm

Range of non-dimensional parameters

τ =h1

h02 to 2.5

c∗ =ch0

0.008 to 0.013

4 Results and Discussion

The effect of roughness on performance parameters like pressure and frictionin slider bearing analysis are very important. Insufficient pressure and muchfriction cause the direct contact and deterioration in the bearing performance.So, here the main focus of the study was to analyze pressure distribution andfriction with respect to various roughness parameters and functions.

Figures 3 and 4 deals with the pressure profile with respect to various types ofsurface roughness patterns. It appears that the pressure is significantly elevatedfor Beta distribution in comparison with Rectangular and Normal distribution.

It is observed that, when the deviation from the mean film thickness isincreased, the pressure is decreased for all roughness patterns. This may be dueto the fact that the in crease in deviation always retards the flow of lubricantwhich might resulted in decrease in pressure.

In the Fig. 6, it is noticed that, when the film thickness ratio is increased,the pressure is decreased for all roughness patterns due to increase in mean filmthickness. It is also found that the effect of the film thickness ratio is nominalin Normal distribution as compared to other two distributions.

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Fig. 3. Non-dimensional pressure distribution E(p∗) versus α for different value of c∗

Fig. 4. Non-dimensional pressure distribution E(p∗) versus α for different value of τ

Fig. 5. Non-dimensional pressure distribution E(p∗) versus c∗ for different value of α

Fig. 6. Non-dimensional pressure distribution E(p∗) versus τ for different value of α

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Tribology 187

Table 2. Comparison of pressure distribution of three types of surface roughness pat-terns

Roughnessheight c∗

E(p∗) ofRectangulardistribution

E(p∗) ofPseudo-normaldistribution

E(p∗) ofBetadistribution

Pes.Increases inE(p∗) ofBetadistributionwithreference toPseudo-normaldistribution

Pes.Increases inE(p∗) ofBetadistributionwithreference toRectangu-lardistribution

0.008 156.6236581 108.3720338 239.4870151 120.985993 52.9060283

0.009 139.1703443 96.28002911 212.8266192 121.0496 52.9252659

0.01 125.2076865 86.60642251 191.4982862 121.113262 52.9445129

0.011 113.7836869 78.69165061 174.0478155 121.176979 52.9637686

0.012 104.2636803 72.09600446 159.50574 121.240748 52.9830325

0.013 96.20828324 66.5150702 147.2008903 121.304570 53.0023042

Table 2 shows the comparison of pressure distribution obtained using differentroughness distribution functions. It is seen that the pressure distribution patternfound nearly similar for all the three functions. Pressure gets decreases withrespect to increase in roughness parameters. It is also clearly found that theBeta distribution with reference to Pseudo-normal distribution gives 120 Pes.more pressure and 53 Pes. more with reference to Rectangular distribution.

It is observed from Figs. 7, 8 and 9 that the coefficient of friction increaseswith increase in surface roughness. Moreover, it is interesting to note that thecoefficient of friction found more for Normal distribution in comparison with Betadistribution and Rectangular distribution. It is also found that the coefficientof friction increases with increase in film thickness ratio. Probably the reasonbehind this is that the fluid friction increases due to increase in film thickness(Table 3).

Fig. 7. Coefficient of friction µ versus α for different value of c∗

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Fig. 8. Coefficient of friction µ versus α for different value of τ

Fig. 9. Coefficient of friction µ versus τ for different value of c∗

Table 3. Comparison of coefficient of friction of three types of surface roughness pat-terns

Roughnessheight c∗

µ ofRectangulardistribution

µ ofPseudo-normaldistribution

µ of Betadistribution

Pes.Decreases inµ of Betadistributionwithreference toPseudonormaldistribution

Pes.Decreases inµ of Betadistributionwithreference toRectangulardistribution

0.008 0.025192686 0.036149257 0.016567764 118.190318 66.13186363

0.009 0.028294925 0.040570231 0.018618672 117.9007756 5.93007952

0.01 0.031386742 0.044969461 0.020665118 117.610471 65.72775782

0.011 0.034468132 0.049346937 0.022707101 117.319407 65.52489882

0.012 0.037539093 0.053702651 0.02474462 117.027581 65.32150279

0.013 0.04059962 0.05803659 0.026777674 116.734995 65.11757001

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Tribology 189

The comparison of coefficient of friction obtained using different roughnessdistribution functions are displayed in Table 2. It is observed that the co efficientof friction gets increase with respect to increase in roughness parameter for all thethree distribution functions. Here, Beta distribution with reference to Pseudo-normal distribution gives 117 Pes. less coefficient of friction and 65 Pes. less withreference to Rectangular distribution.

The graphical as well as tabular representation shows that the proper selec-tion of design parameters as well as theoretical roughness distribution functionmay help in better design of bearing system.

5 Conclusion

It is needless to say that the film thickness ratio and roughness parameters mayplay an important role in providing a better performance of bearing system. So,the careful selection of the design parameters can significantly improve the per-formance characteristics. Besides, the bearing performance is improved in Betadistribution as compared to rectangular and normal distribution. The significantrise in pressure distribution and drop in coefficient of friction has been foundin case of Beta distribution as compared to other two distribution functions.These outcomes suggest that the proper use of distribution function may lead tothe exact evaluation of the roughness effect on the performance of the bearingsystem. So, based on the present analysis it can be concluded that the properchoice of distribution function may become helpful to the engineers for betterdesign of the bearing systems.

References

1. Bhushan, B. (ed.): Handbook of Micro/Nano Tribology. CRC Press, New York(1995)

2. Tzeng, S.T., Saibel, E.: Surface roughness effect on slider bearing lubrication.Trans. ASME. J. Lub. Tech., 10, 334-338, (1967a)

3. Christensen, H., Tonder, K.C.: Tribology of Rough Surfaces: Stochastic Models ofHydrodynamic Lubrication. SINTEF Report No. 10/69-18 (1969a)

4. Christensen, H., Tonder, K.C.: Tribology of Rough Surfaces: Parametric Study andComparison of Lubrication Models. SINTEF Report No. 22/69-18 (1969b)

5. Christensen, H., Tonder, K.C.: The hydrodynamic lubrication of rough bearingsurfaces of finite width. ASME-ASLE Lubrication Conference; Paper No. 70-Lub-7 (1970)

6. Prakash, J., Tiwari, K.: Roughness effects in porous circular squeeze-plates witharbitrary wall thickness. J. Lub. Tech. 105, 90–98 (1983)

7. Patel, H.C., Deheri, G.M.: Characteristics of lubrication at nano scale on the per-formance of transversely rough slider bearing. Mechanics 80(6), 64–71 (2009)

8. Sinha, P., Adamu, G.: THD analysis for slider bearing with roughness: specialreference to load generation in parallel sliders. Acta mechanica 207(1-2), 11–27(2009)

9. Gururajan, K., Prakash, J.: Surface roughness effects in infinitely long porous jour-nal bearings. J. Tribol. 121(1), 139–147 (1999)

Publication 1 105


10. Panchal, G.C., Patel, H.C., Deheri, G.M.: Influence of Surface Roughness througha series of flow factors on the performance of a longitudinally rough finite sliderbearing. Ann. Fac. Eng. Hunedoara 14(2), 227 (2016)

11. Patel, N.S., Vakharia, D., Deheri, G.: Hydrodynamic journal bearing lubricatedwith a ferrofluid. Ind. Lubr. Tribolo. 69(5), 754–760 (2017)

12. Ohkubo, T., Kishigami, J.: Accurate measurement of gas-lubricated slider bearingseparation using visible laser interferometry. J. Tribol. 110(1), 148–155 (1988)

Publication 1 106

Journal of the Serbian Society for Computational Mechanics / Vol. 9 / No. 2, 2015 / pp. 10-18

(UDC: 621.822.5:532.54)

Magnetic Fluid Lubrication of an Infinitely Long Slider Bearing with Rough Surfaces

M. P. Patel1*, Dr. H. C. Patel2, Dr. G. M. Deheri3

1General Department, K. D. Polytechnic, Patan, Gujarat, India. [email protected] 2Gujarat University, Ahmedabad, Gujarat, India. [email protected] 3Department of Mathematics, V.V. Nagar, Anand, Gujarat, India. [email protected] * Corresponding author

Abstract

This article deals with the performance characteristics of an infinite slider bearing with rough surfaces in the presence of a magnetic fluid as the lubricant. The magnetic fluid flow model of Neuringer Rosenwicg has been used here. The stochastic model of Christensen and Tonder has been adopted to evaluate the effect of surface roughness. The related stochastically averaged Reynolds equation is solved to obtain the pressure distribution leading to the calculation of load carrying capacity. The results presented in graphical forms suggest that the adverse effect of transverse roughness can be minimized by choosing suitable magnetic strength. It is observed that the negativity-skewed roughness plays an important role in improving the bearing performance.

Keywords: Slider bearing, magneticfluid, roughness, loads carrying capacity.

1. Introduction

Fluids with strong magnetic properties have attracted considerable attention during the last decade. The use of the magnetic fluid as a lubricant for the bearing system in technical applications in the domain of nano scale science and technology has made significant progress. Magnetic fluid consists of colloidal magnetic nano particles dispersed with the aid of surfactants in a carrier liquid. In reality, magnetic fluid is a hybrid of soft material and the nano particles. The average diameter of the dispersed particles ranges from 3 to 10 nm. The ferrofluids contain enormous magnetic nanoparticles and, therefore, can be influenced by either parallel or perpendicular magnetic field. The use of magnetic fluid has resulted in the development of many energy devices and instruments. Computer disks drives, semiconductors and high precision speakers are commercially available and based on magnetic fluid effects. The most important property of a magnetic fluid is that it can be retained at a desired location under the magnetic field. When a magnetic field is applied, each and every particle experiences a body force causing it to flow.

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In most of the studies conventional lubricants were used. The use of magnetic fluid as a lubricant modifying the performance of the bearings has been well established. The application of magnetic fluid as a lubricant was investigated by many authors (Agrawal (1986), Bhat and Deheri (1995), Odenbach (2004), Nada and Osman (2007), Urreta et al. (2009), Huang et al. (2011)). In all these studies it has been established that the performance of bearing system could be improved by using a magnetic fluid as the lubricant.

In the field of engineering and technology, slider bearings are widely used because of their stability, durability and high load carrying capacity. The contribution of surface roughness and properties of lubricant film on the load carrying capacity and friction is an important aspect in the analysis of slider bearings. The researchers have carried out the study of performance characteristics of slider bearings with different lubricants from time to time. Surface roughness evaluation is very important for many fundamental problems such as friction, load carrying capacity, contact deformation, heat and electric current conditions. For this reason, surface has been the subject of experimental and theoretical investigations for many decades. In literature, many investigations, such as Tzeng and Saibel (1967), Christensen and Tonder (1969a, 1969b, 1970), Gupta and Deheri (1996), accounting for surface roughness effect, have been proposed in order to seek a more realistic representation of bearing surfaces. Patel and Deheri (2009) analyzed the characteristics of lubrication at nano scale on the performance of transversely rough slider bearing. Sinha and Adamu (2009) studied the thermal and roughness effects on different characteristics of an infinite tilted pad slider bearing. In this paper roughness was assumed to be stochastic, and the method was developed using the models of Christensen and Tonder (1969a, 1969b, 1970). Lin et al. (2004) analyzed the squeeze‐film performance between curved annular plates with an electrically conducting fluid in the presence of a transverse magnetic field. Deheri et al. (2005) discussed the effect of transverse roughness on the behavior of slider bearings with squeeze film formed by a magnetic fluid. Lin et al. (2006) observed the effects of fluid inertia on the squeeze film behavior between two parallel annular disks with an electrically conducting fluid in the presence of a transverse magnetic field. Shimpi and Deheri (2012) dealt with the performance of a magnetic fluid based rough short bearing incorporating deformation effect. Deheri et al. (2013) investigated the performance characteristics of a hydrodynamic long journal bearing taking recourse to a magnetic fluid lubricant. Patel and Deheri (2014) analyzed the combined effect of roughness and slip velocity on the performance of a ferrofluid based rough porous hyperbolic slider bearing.

Here it has been thought proper to discuss the performance characteristics of an infinitely long slider bearing with rough surfaces under the presence of a magnetic fluid.

2. Analysis

The configuration of the bearing, which is infinite in the Z direction, is presented in Fig. 1. It consists of two rough surfaces separated by a lubricant film. The lower surface is moving with the uniform velocity U in the X direction, while the upper surface is stationary. The length of the bearing is L while h0 and h1 being the minimum and maximum film thickness. The bearing surfaces are assumed to be transversely rough.

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Fig. 1. Following the discussions of Christensen and Tonder the stochastic film thickness is taken

as:

( ) ( ) ( )

where h is the nominal (smooth) part which measures the large scale part of the film geometry including any long wave length disturbances, and δ is a randomly varying quantity with zero mean, which arises due to the surface roughness measured from the nominal level. In this theory, the film thickness is assumed to be of such a form that the application of the Reynolds equation remains valid. This requirement is basic and quite unconnected with the idea of viewing film thickness as a stochastic process. Christensen and Tonder (1969a) and Christensen (1969) have made the following assumptions:

1. The magnitude of the pressure ripples associated with the surface roughness is small compared to the general pressure level in the bearing, and consequently, the variance of the pressure gradient in the roughness direction is negligible.

2. In the direction perpendicular to the roughness direction, the variance of unit flow is negligible.

3. The magnitudes of temperature and velocities associated with roughness are small compared to the corresponding general magnitudes in the bearing.

The details discussion regarding this aspect can be found in the investigation of Christensen and Tonder (1969a) and Christensen (1969).

Following the averaging method of Christensen and Tonder the roughness turns out to be:

Following the investigations of Agrawal (1986) and Bhat (2003) the magnitude H of the magnetic field is taken as:

( )

K being a quantity chosen to suit the dimensions so as to manufacture a magnetic field of required strength.

In view of usual assumptions of hydrodynamic lubrications, the lubricant film is taken to be isoviscous, incompressible, and the flow is laminar. Following the method of Sinha (2009), Agrawal (1986) and Deheri (2014) and adopting the stochastic averaging by the method of Christensen and Tonder, the Reynolds equation governing the pressure distribution is obtained as:

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( ( )

(

))

( ( ))

where

( )

( )


Introduction of dimensionless quantities

( )

( )

leads to the dimensionless Reynolds equation:

( ( )

(

( )))

Integrating both sides one finds that

(

( ))

( )

where subscript m refers to the condition at point where

.

The associated boundary conditions are

Then, the expression for nondimensional pressure distribution is found to be:

( ) ( ) ∫

( )

where

∫ ( )

∫ ( )

Lastly, the dimensionless load-carrying capacity is obtained from the relation:

∫ ( )

3. Results and discussions

It is clearly seen that the equation:

( ) ( ) ∫

( )

determines the pressure distribution while the load carrying capacity is obtained from:

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∫ ( )

Both these equations depend on various parameters such as . The first parameter describes the effect of magnetization while the remaining three parameters decide effect of roughness.

Fig. 2. indicates that load carrying capacity decreases with increased standard deviation, which does not happen in case of longitudinal roughness. Besides, variance (+Ve) decreases the load

carrying capacity while the load carrying capacity gets increased due to variance (-Ve).

Fig. 3. The fact that the trends of load carrying capacity with respect to skewedness are similar to that of variance is reflected in Fig.3. Of course, the effect of skewedness on the load carrying

capacity with respect to variance is nominal.

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Fig. 4. displays that the positively skewed roughness compounds the negative effect of standard deviation.

Fig. 5. suggests that the effect of skewness on the variation of load carrying capacity with respect to the film thickness ratio is almost nominal. It is established that the positive effect of variance (-ve) gets enhanced by the negatively skewed roughness, which may go a long way in

improving the performance of the bearing system.

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ζ* μ*=0.1 μ*=0.2 μ*=0.3 μ*=0.4 μ*=0.5 0 0.051062318 0.059395651 0.067728985 0.076062318 0.084395651 0.05 0.05101616 0.059349493 0.067682827 0.07601616 0.084349493 0.1 0.050878055 0.059211388 0.067544722 0.07601616 0.084211388 0.15 0.050649185 0.058982518 0.067315852 0.075649185 0.083982518 0.2 0.050331731 0.058665064 0.066998397 0.075331731 0.083665064

ε* -0.05 0.050977085 0.059310419 0.067643752 0.075977085 0.084310419 -0.025 0.050928081 0.059261414 0.067594748 0.075928081 0.084261414 0 0.050878055 0.059211388 0.067544722 0.075878055 0.084211388 0.025 0.050827103 0.059160436 0.067493769 0.075827103 0.084160436 -0.05 0.050775309 0.059108643 0.067441976 0.075775309 0.084108643

α* -0.05 0.096061181 0.147061181 0.198061181 0.249061181 0.300061181 -0.025 0.09477764 0.14577764 0.19677764 0.24777764 0.29877764 0 0.093544722 0.144544722 0.195544722 0.246544722 0.297544722 0.025 0.092359836 0.143359836 0.194359836 0.245359836 0.29635986 -0.05 0.09122056 0.14222056 0.19322056 0.24422056 0.29522056

Table 1. Variation in Load carrying capacity with respect to ζ*, ε*, α*

From Table 1 it is concluded that the load carrying capacity sharply rises due to magnetization, which may be due to the fact that the viscosity of the lubricant increases owing to the magnetization. It is interesting to note that for this type of bearing, system magnetization does not allow the load carrying capacity to be affected much due to roughness parameters.

4. Conclusion

Although the effect of negatively skewed roughness remains positive while designing the bearing system, the roughness aspects must be evaluated. For this type of bearing system the bearing may turn in an enhanced performance irrespective of the fact that the machine has run for a long time.

Извод

Лубрикација магнетног флуида код бесконачног клизног лежаја храпаве површине

M. P. Patel1*, Dr. H. C. Patel2, Dr. G. M. Deheri3

1General Department, K. D. Polytechnic, Patan, Gujarat, India. [email protected] 2Gujarat University, Ahmedabad, Gujarat, India. [email protected] 3Department of Mathematics, V.V. Nagar, Anand, Gujarat, India. [email protected] * Corresponding author

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Резиме

У овом раду се анализирају карактеристике перформанси бесконачног клизног лежаја храпаве површине у присуству магнетног флуида као лубриканта. Коришћен је Neuringer Rosenwicg модел протока магнетног флуида. За процену утицаја храпавости површине примењен је Christensen и Tonder-ов стохастички модел. Одговарајућа стохастички усредњена једначина Рејнолдсовог типа је решена да би се добила дистрибуција притиска и након тога израчунао капацитет носивости. Резултати приказани графички показују да се нежељени утицај трансферзалне храпавости може свести на минимум уколико се одабере одговарајућа снага магнета. Запажа се да негативно закошена храпава површина има важну улогу у побољшању перформанси лежаја.

Кључне речи: клизни лежај, магнетни флуид, храпавост, капацитет носивости.

References

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Patel SJ, Deheri GM, Patel JR(2014). Ferrofluid Lubrication of a Rough Porous Hyperbolic Slider Bearing with Slip Velocity. Tribology in Industry. 36, 3, 259-268.

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