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## C O N T E N T S

ACKNOWLEDGMENTS 6

PREFACE 8

CHAPTER – 1 GENERAL INTRODUCTION 15

1.1 GENERAL INTRODUCTION: 16

1.2 TYPES OF LUBRICANTS: 18

1.2.1 Fluid-lubricant Properties: 22

1.3 MODE OF OPERATION: 24

1.3.1 Hydrodynamic Lubrication: 24

1.3.2 Hydrostatic Lubrication: 25

1.3.3 Boundary Lubrication: 26

1.3.4 Elastohydrodynamic Lubrication: 27

1.3.5 Partial (Mixed) Lubrication: 28

1.3.6 Turbulent Lubrication Regime: 29

1.3.7 Magnetohydrodynamic Lubrication (MHD): 29

1.3.8 Rarefied Gas Lubrication: 30

1.3.9 Porous Metal Lubrication: 31

1.3.10 Biolubrication: 32

1.4 TYPES OF RELATIVE MOTION: 32

1.5 GEOMETRY OF BEARING SURFACES: 33

1.6 TYPES OF LOADING: 34

1.7 BEARING DESIGN CHARACTERISTICS: 34

1.8 REVIEW OF RELATED LITERATURE: 35

CHAPTER – 2 RELATED ASPECTS AND DERIVATION OF MODIFIED REYNOLDS EQUATION

46

2.1 INTRODUCTION: 47

2.1.1 Inertia and Turbulent Effects in Lubrication: 48

2.1.2 Thermal Effects in Lubrication: 49

2.1.3 Non-Newtonian Lubricants: 51

2.1.4 Surface Roughness Effects in Lubrication: 52

2.2 MATHEMATICAL MODELLING OF A BEARING SYSTEM: 53

2.2.1 Equation of State: 55

2.2.2 Constitutive Equation of the Lubricant: 56

2.2.3 Continuity Equation: 58

2.2.4 The Equations of Motion: 59

4

2.2.5 Energy Equation: 61

2.2.6 Elastic Considerations: 62

2.2.7 Surface Roughness: 62

2.3 BOUNDARY CONDITIONS: 64

2.4 BASIC ASSUMPTIONS OF HYDRODYNAMIC LUBRICATION: 65

2.5 MODIFIED REYNOLDS EQUATION: 67

2.6 THE BEARING PERFORMANCE CHARACTERISTICS: 73

CHAPTER – 3 EFFECT OF SURFACE ROUGHNESS ON THE HYDRODYNAMIC LUBRICATION

OF SLIDER BEARINGS 74

3.1 INTRODUCTION: 75

3.2 TRANSVERSE SURFACE ROUGHNESS: 77

3.2.1 Analysis: 77

3.2.2 Results and Discussions: 82

3.3 LONGITUDINAL SURFACE ROUGHNESS: 104

3.3.1 Analysis: 104

3.3.2 Results and Discussions: 108

CHAPTER – 4 ON THE OPTIMUM FILM PROFILE 122

4.1 INTRODUCTION: 123

4.2 TRANSVERSE SURFACE ROUGHNESS: 124

4.2.1 Analysis: 124

4.2.2 RESULTS AND DISCUSSIONS: 130

4.3 LONGITUDINAL SURFACE ROUGHNESS: 135

4.3.1 Analysis: 135

4.3.2 Results and Discussions: 139

CHAPTER – 5 EFFECT OF SURFACE ROUGHNESS ON THE BEHAVIOUR OF SQUEEZE FILM

BEARING 144

5.1 SQUEEZE FILM IN A SPHERICAL BEARING: INTRODUCTION: 145

5.2 TRANSVERSE SURFACE ROUGHNESS OF A SPHERICAL BEARING: 147

5.2.1 Analysis: 147

5.2.2 Results and Discussions: 151

5.3 LONGITUDINAL SURFACE ROUGHNESS OF A SPHERICAL BEARING: 162

5.3.1 Analysis: 162

5.3.2 Results and Discussions: 165

5.4 SLIDER BEARINGS WITH SQUEEZE FILM FORMED BY A MAGNETIC FLUID:

INTRODUCTION: 177

5.5 TRANSVERSELY ROUGH SLIDER BEARING WITH MAGNETIC FLUID: 178

5

5.5.1 Analysis: 178

5.5.2 Results and Discussions: 180

5.6 LONGITUDINALLY ROUGH SLIDER BEARING WITH MAGNETIC FLUID: 198

5.6.1 Analysis: 198

5.6.2 Results and Discussions: 200

CHAPTER – 6 GENERAL CONCLUSIONS AND FURTHER SCOPE FOR INVESTIGATIONS 218

6.1 GENERAL CONCLUSIONS: 219

6.2 FURTHER SCOPE FOR INVESTIGATIONS: 222

REFERENCES 224

N O M E N C L A T U R E 243

## ACKNOWLEDGMENTS

I wish to express my profound sense of gratitude to (Retd.) Prof. J. L

Gupta, Professor and Head, Department of Mathematics, B.V.M. Engineering College, Vallabh Vidyanagar and Dr. G. M. Deheri, Reader, Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar, who have been unsparing in their efforts to provide me with a very valuable guidance and generous encouragement throughout the course of my present investigation and thesis preparation

I am thankful to Prof. M.D. Patel, Head, Department of Mathematics, Sardar Patel University for providing necessary facilities to carry out my work. I am also thankful to all the teaching and non-teaching staff members of the Mathematics Department, Sardar Patel University, Vallabh Vidyanagar, who have helped me directly or indirectly in completing this work

I thank Prof. B. P. Swadas, Principal, B.V.M. Engineering college, (Retd.) Prof. M.H. Vasavda, Ex-Head, Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar for their co-operation during my work of this thesis

Above all, how can I forget to express my special thanks to Prof. B.K. Oza, Vice-Chancellor, Shri Bharatsinhjibhai Parmar, Registrar, (Retd.) Prof. J.K. Rao, Ex-Head, Department of Mathematics, Prof. G.M. Akolia, Head, Department of Mathematics, friends and colleagues from Bhavnagar University, Bhavnagar, who have been very co-operative in my progress

Especially, I would like to thank my wife Priti and other family members, family members of my investigators-in-charge, Dr. A.H. Hasmani and his family members, who smilingly tolerated all the difficulties while I was engrossed in working for this dissertation. I put on record sincere thanks for Mr. G.R

Andharia, Mr. R.I. Andharia and Ms. Kajal Dave for helping me in the various stages of this thesis

P. I. Andharia Department of Mathematics, Sardar Patel University, October 23, 2000 Vallabh Vidyanagar

## PREFACE

The word "Tribology" was introduced when the "Department of Education and Science Report", known as "Jost Report" published in England in 1966. Tribology is defined as the science and technology of interacting surfaces in relative motion. It is an integrated study of lubrication, friction and wear of moving or stationary parts. It is the friction developed due to the interaction between opposing surfaces, resists the relative motion of the surfaces. Friction causes unsmooth relative motion of the surfaces and wear. It may result in reduction of the machine life. Therefore, various investigations have been launched to minimize this friction. In order to minimize friction and wear between moving machine elements a foreign substance, known as lubricant is introduced in between them. This lubricant keeps the machine elements apart and allows them to slide on each other with minimum efforts

Presence of a lubricant film between two mating parts greatly reduces surface wear and consequently deformation, loss of energy, expansion of material by local surface heat, seizure of surfaces, unsmooth relative motion and maintenance or running cost of machines. When two surfaces of a machine are moving relative to each other, their lubrication depends upon a number of factors like load on the surfaces, relative velocity of the surfaces, geometry of the surfaces, the type of materials out of which the surfaces are made and physical and chemical properties of the lubricants etc

The contamination of lubricant is basically responsible for making a bearing surfaces rough through chemical degradation. In some cases bearing surfaces after having some run-in and wear develop roughness. Surface roughness of the bearing significantly affects the bearing performance. Hence the surface roughness has been a subject of discussion in many recent investigations

The roughness has three different structures: (1) Transverse roughness: a 1-dimensional disturbance with the furrows across the sliding direction (2) Longitudinal roughness: a 1-dimensional disturbance consisting of grooves and ridges in the sliding/rolling direction (3) Uniform roughness: a 2-dimensional disturbance distributed uniformly over the surface

The effect of roughness on the performance of the bearing may be different for different structures. The present dissertation is centered around the study of the effect of surface roughness (i.e. transverse and longitudinal) on the performance of the bearing system

The content of this dissertation is designed in six chapters. The first chapter is introductory in nature. It deals with the role of the bearings in machines and the factors that are important for the process of lubrication; such as types of lubricants, types of bearings, geometry of surfaces, types of loading etc. It follows by a discussion on bearing design characteristics. Lastly, the review of the related literature is presented in this chapter

In the second chapter of this dissertation mathematical modelling of the bearing system is discussed in details. This chapter also deals with the effects of various factors like inertia, turbulent, thermal and surface roughness in lubrication. Efforts have been 10 made to focus particularly, on the surface roughness effects. Besides, the modified Reynolds equation for incompressible fluid is obtained in this chapter

The third chapter presents the surface roughness effect on hydrodynamic lubrication of slider bearings. This chapter consists of two sections. The first section is concerned with the analysis of one-dimensional slider bearing with transverse surface roughness. The second section considers one-dimensional slider bearing with longitudinal surface roughness. In these investigations the probability density function for the random variable characterizing the surface roughness is considered to be assymetrical with non-zero mean. In both the cases four different lubricant film shapes such as plane slider, exponential, secant and hyperbolic are taken into consideration

Results are numerically computed and presented graphically as well as in tabular form

The analyses discuss the variation of the pressure and the performance characteristics such as load carrying capacity, frictional force, centre of pressure and temperature rise for different values of roughness parameters. It is found that all the three roughness parameters affect the bearing performance characteristics; however, the effect of the parameter describing symmetricity seems more sharp. While in the case of transverse surface roughness the bearing suffers mostly, sometimes negatively skewed roughness marginally betters the performance. It is interesting to note that sometimes longitudinal roughness enhances the performance of the bearing

The fourth chapter describes the analysis of an optimum film profile of a rough slider bearing. The study has been made for both the one-dimensional structures of roughness; namely, transverse and longitudinal. An attempt has been made to find the shape of the lubricant film profile for a rough slider bearing such that the load carrying capacity of the bearing is optimum (maximum). It is observed that the optimum film profile for such a bearing is a step function. All the three parameters characterizing the roughness are taken into consideration. The step location and the step height ratio are calculated for various values of roughness parameters, and in turn, these results are used to compute load carrying capacity, frictional force and the centre of pressure. All the results are presented in tabular form and graphically as well. The results show that negatively skewed asymmetric surface roughness enhances the performance

In the fifth chapter effect of surface roughness on the behaviour of squeeze film spherical bearing is analysed. Both the structures of roughness; transverse and longitudinal, are considered. In these discussions the probability density function for the random variable characterizing the surface roughness is taken to be symmetric with non-zero mean. The study reveals that while mostly the surface roughness adversely affects the load carrying capacity of the bearing and causes the reduction of the response time for the rotor to attain a given film thickness; the standard deviation increases the load carrying capacity in the case of longitudinal surface roughness

Towards the second half of this chapter the effect of surface roughness on the behaviour of slider bearing with squeeze film formed by a magnetic fluid is discussed

Both the structures, namely: transverse and longitudinal are taken into consideration

The findings say that the use of magnetic fluid as lubricant can reduce the adverse effect of the surface roughness in the performance of the bearing

In the last chapter of this dissertation a brief summary and general conclusions of the present study are outlined. The scope for further investigations and future works is also suggested in the same chapter

The findings of this dissertation tend to suggest that there is a possibility of the load carrying capacity being increased in the case of longitudinal roughness while the bearing suffers mostly in the case of transverse roughness. Further, the use of magnetic fluid as lubricant can reduce the adverse impact of the roughness on slider

The investigation carried out in this dissertation establishes firmly that the effect of roughness on the performance characteristics is considerably significant and hence it must be accounted for in the bearing design

Substantial portion of the work incorporated in this dissertation has been published in various research journals of national and international repute and also presented at national and international conferences. One of the article was adjudged to be the best research paper in Gujarat Science Academy. Another paper was chosen for Hari Ohm Ashram award for the year 1997 - 98 in the field of Mechanical Engineering. Details of the papers incorporated in this thesis are listed below: 1. Effect of Longitudinal Surface Roughness on Hydrodynamic Lubrication of Slider Bearings, Proceeding of SMT-X, Published by The Institute of Materials, London, U.K., (1997) 872-880

This paper was presented at 10th International Conference on Surface Modification Technologies held in Singapore Sept. 2-4, 1996

This paper was also chosen for Hari Ohm Ashram award for the year 1997- 98

(Incorporated here as a part of section 3.3) 2. Effect of Longitudinal Surface Roughness on Hydrodynamic Lubrication of Hyperbolic Slider Bearings, National Academy of Sciences, India, Allahabad, Vol. 68, Section A, Part III, (1998) 249-256

(Incorporated here as a part of section 3.3) 3. Optimum Film Profile of a Slider Bearing with Longitudinal Surface Roughness, 'Prajna'- Journal of Sardar Patel University, Vol. 8, (1998) 23-28

Ø This paper was presented at 12th Gujarat Science Congress held on 15-16 February, 1997 at Vallabh Vidyanagar

(Incorporated here as section 4.3) 4. Effect of Transverse Surface Roughness on the Behaviour of Squeeze Film in a Spherical Bearing, Proceedings of the International Conference : Problems of Non-conventional Bearing Systems, NCBS '99, Zielona Gora, 15-18 Sept., 1999; Applied Mechanics and Engineering, (1999) Vol. 4, Special Issue : NCBS '99, 19-24

Ø This paper was presented at 14th Gujarat Science Congress held on 10-11 October, 1998 at Palitana. This paper was selected for the award of Best Research Paper

(Incorporated here as section 5.2) 5. Optimum Film Profile of a Slider Bearing with Transverse Surface Roughness, 'Prajna'- Journal of Sardar Patel University, Vol. 9, (1999) 22-28

Ø This paper was presented at 12th Gujarat Science Congress held on 15-16 February, 1997 at Vallabh Vidyanagar

(Incorporated here as section 4.2) 6. Effect of Surface Roughness on Hydrodynamic Lubrication of Slider Bearings, accepted for publication in Tribology Transactions (U.S.A.)

(Incorporated here as section 3.2) 7. On the Shape of the Lubricant Film for the Optimum Performance of a Longitudinal Rough Slider Bearing, accepted for publication in Industrial Lubrication Tribology (West Yorkshire, U.K.)

(Incorporated here as section 4.3) 8. Effect of Longitudinal Surface Roughness on the Behaviour of Squeeze Film in a Spherical Bearing, communicated to Applied Mechanics and Engineering, Poland

(Incorporated here as section 5.3)

## CHAPTER – 1

## GENERAL INTRODUCTION

## 1.1 GENERAL INTRODUCTION:

The present day machine technology is dependent on mechanisms involving kinetic pairs where mechanical power has to be transmitted between surfaces that are in relative motion. Two most important inherent phenomena associated with such a system are those of friction and wear. It is the friction which resists relative motion of the surfaces and causes wear. It consumes and wastes energy due to noise and generation of local surface heat. Wear causes changes in dimensions and results in eventual breakdown of the machine element and consequently the entire machine and all that depend upon it. The aspect of loss of energy and loss of material due to friction is astonishingly large

Professor Vogelpohl (1951) [Hamrock (1994)] has estimated that from one-third to one-half of total energy produced in the world is consumed in friction. Automobiles, trucks, buses, trains, airplanes, ships and the like, effectively expenses most of their power in overcoming friction. The automobile engine delivers useful work only after the friction is reduced. This useful work is then largely consumed in gear friction, rolling friction of the tires, brake friction, and wind friction etc

The effect of wear is of equally momentous significance. Wear may be defined as deterioration of surface due to use. It occurs in wide variety of operations and in some industries the annual cost of replacing worn parts is a major expense. Wear causes the deformation of the machine elements and changes in its dimensions which may result in seizure of surfaces, reduction in machine life or in its serious break down thus enormously increasing the running and maintenance cost of machines. For the conservation of energy and material from natural resources point of view the reduction of friction and wear through proper means is a basic necessity

In 1966 with the publication in England of the "Department of Education and Science Report", some times known as the "Jost Report", the word "Tribology" was introduced and defined as the science and technology of interacting surfaces in relative motion and of the practice related there to. A better definition of "Tribology" might be "The integrated study of friction, wear and lubrication"

In order to minimize friction and wear between moving machine elements a foreign substance known as lubricant is introduced in between them. The lubricant keeps the machine elements apart and allows them to move relatively with minimum efforts. Such a system where lubricant is used is called lubricated system and the process of minimizing the friction and wear using lubricant is called lubrication. The machine component which serves to achieve the smooth motion of the machine surfaces using lubricant is called bearing. Bearing is an important component in machines. It is a support or guide that locates one component with respect to others in such a way that prescribed relative motion can occur while forces associated with machine operation are transmitted smoothly and efficiently. It transmits loads and forces or supports load exerted on the moving machine parts under reduced friction. Design of bearing requirements are imposed on the mechanical system by the bearings. There is a continuous demand for better bearing systems to support moving machine components in regard to improved performances and exacting operating conditions and research efforts are directed towards achieving these objectives

Bearings may primarily be classified on the basis of following factors :

(1) types of lubricants

(2) mode of operation

(3) types of relative motion

(4) geometry of bearing surfaces

(5) types of loading

(6) type of bearing materials.

### 1.2 TYPES OF LUBRICANTS:

Selection of lubricant depends upon the type of the bearing which is used in the machine elements. Solids, liquids, gases or even plasma are used as lubricants

Commercial lubricants can roughly be grouped in three generic type namely; solid lubricants, semi-solid lubricants and liquid lubricants. The bearings using solid lubricants are called dry rubbing bearings, whereas those using fluid as lubricants are called conformal fluid film bearings. In dry rubbing bearings the load carrying and frictional characteristics can directly be related to basic contact properties of the bearing materials and the lubricant used

Solid lubricants are used in cases where lubricating oils and greases can not be used because of contamination reasons. Solid lubricants are used in high temperature conditions or in machines working under high pressure and low speed condition

Common semi-solid lubricant are greases and polymer thickened oils. Grease consists of a soap dispersed throughout a liquid lubricating oil. Main function of the soap is of thickening agent so that grease sticks firmly to bearing surface. These are used in variety of applications like rail axle boxes, paper textile and food producing, machinery and bearings and gears working under high temperatures like hot rolling machine where it is used to reduce roll load

Most liquid oil lubricants can be classified in to vegetable oils, mineral or petroleum oil, blended oil, synthetic oil and emulsions

Animal and vegetable oils have good oiliness even under high temperatures and loads. They undergo oxidation and have tendency to hydrolyze in moist air and are used as blending agents with other lubricating oils. They are mainly used for delicate instruments like watches, clocks, sewing machines etc

Mineral or petroleum oils are obtained by distillation of petroleum. These types of lubricants are stable under service conditions but possess poor oiliness. Naphthenic and paraffinic are the main two types of such oil. The first kind acts as a very good lubricant for almost any application. It contains very little wax, where as the paraffinic oil is very much waxy and is used in hydraulic equipment, steam engines, elevators, cranes etc

It is possible to improve the typical lubricating properties of petroleum oils by adding specific materials as additives. Oiliness of a lubricating oil can be increased by the addition of vegetable oils and fatty acids. In many applications the lubricant becomes mixed with air as is frequently the case. It will begin to oxidize and the rate of oxidation will get accelerated if catalytic action of metals is present and as the temperature rises. This may produce unwanted products - the gums, varnishes and the sludge and the lubrication is seriously interfered with

Synthetic lubricants can extend the life of equipment which operates at extreme temperatures : some eliminate fires of oxidation with highly reactive materials and they reduce maintenance and down time. Such lubricants have remained special lubricants for unusual applications. The important synthetic lubricants are silicon fluids. Most widely recognized synthetic lubricants are silicones having least change in viscosity, wide temperature range, good thermal stability and good oxidation stability

An emulsion is two phase system consisting of a fairly coarse dispersion of two immiscible liquids, one being dispersed as droplets in other. Oil in-water or water in-oil emulsions are used in several situations as lubricants. 3 to 20 % water emulsions are used for cutting tools, 40 % water emulsions are used for compressors and pneumatic tools. High viscosity oil emulsions with water are used for steam engines. Soap emulsions are used for wet wire drawing

Ferroliquids are ordinary liquids containing stable colloidal dispersions of ferrite particles. They behave as ordinary liquids except that they experience a body force in the presence of an applied magnetic field gradient. The use of ferromagnetic fluids as lubricant in fluid film bearing is mainly attracted by its magnetic sealing effect and boundary lubricating effect. Under a suitable design of the bearing geometry and magnetic field, the ferrofluid lubricated bearing can operate without side leakage, so that mechanical seals can be eliminated, and bearing friction at both ends can be reduced. The boundary lubricating effect of the ferrofluid, retained at the sliding surface by magnetic field, can reduce the friction and wear between the bearing surfaces under low velocity operating conditions. [Mikaye and Takahashi (1985)]

Gases are preferred as lubricants on account of its several advantages, like : (I) Low frictional force and torque due to reduced lubricant viscosity

(II) Low power loss, cool running characteristics and very low wear rate

(III) Operate over a large range of speed and temperature (IV) Almost no periodic maintenance, virtually no problem of contamination and therefore no requirement of seal

However, a gas lubricated bearing has inferior load carrying capacity compared to an identical oil bearing, besides they are prone to instability. Air is used as lubricant in many high speed spindles. Wind tunnel balances, torque - meter, machine tool side ways, gyroscopes, accelerometers, electric motors, refrigerators, liquefies, computer elements, dental drill machine etc., use air as lubricant because of the advantage of low frictional force. Gas circulators use gas as lubricant because of its being contamination free. In a number of chemical plants air is used as lubricant because of high temperature conditions

Gas bearings are particularly valuable when used with precision instruments because of low noise characteristics and low frictional losses. When the distance between molecules of the gaseous lubricant becomes large enough the fluid no longer acts as if it were continuum, particularly, in the situations where very small film thickness are involved as in low vacuum conditions such as magnetic storage devices and gyroscopes etc. The lubricant in these situations is a rarefied gas [Ramanaiah (1969)]

In addition to their function of reducing friction the lubricants also perform the function of carrying away a major portion of heat generated by the friction. Thus the theory of lubrication deals not only with the ways of minimizing the friction and wear, but the viscous dissipation of heat also

Most of the lubricants used come from the range of fluids, hence the study of fluid film lubrication has assumed considerable importance

### 1.2.1 Fluid-lubricant Properties:

The lubricant is required to possess certain physical, chemical and metallurgical properties commensurate with the system where it is required to be used

In order that a lubricant is to be effective, it must be viscous enough to maintain a lubricant film under operating conditions but should be as fluid as possible to remove heat and to avoid power loss due to viscous drag. A lubricant should also be stable under thermal and oxidation stresses and have low volatility

This is the most important single property of fluid lubricant as it determines the frictional power loss and heat generation in bearing and the flow rate through the bearing. In general, however, a lubricant does not simply assume a uniform viscosity in a given bearing. This results form the nonuniformity of the pressure and/or the temperature prevailing in the lubricant film. Indeed, many elastohydrodynamically lubricated machine elements operate over ranges of pressure and/or temperature so extensive that the consequent variations in the viscosity of the lubricant may become substantial and in turn, may dominate the operating characteristics of the machine element

It is a well-known fact that many common lubricants, including petroleum - base lubricants, undergo considerable increase in viscosity when they are subjected to high pressures. The increase in viscosity due to pressure will raise the friction losses in the bearing

The viscosity of a lubricating oil decrease with a rise in temperature. In many important practical applications this variation in viscosity with temperature is very significant as is in the case of automobile engine crank-case. High viscosity of oil means excessive bearing resistance during cold starting with heavy demands upon the battery

After the engine has warmed up, the oil viscosity becomes lower. If this reduction in viscosity is too great, wear, or even seizure may then result. Hence knowledge at the variation of viscosity with temperature is of great importance in determining the suitability of a lubricating oil for a particular use

Besides, the variations of viscosity with temperature, a number of other thermal properties of fluids as lubricants are important especially, specific heat and thermal conductivity

Viscosity is sensitive to large variations in both pressure and temperature. This sensitivity forms a considerable obstacle to the analytical description of the consequent viscosity changes. Roelands (1966) noted that at constant pressure the viscosity increases more or less exponentially with the reciprocal of absolute temperature

Similarly, at constant temperature the viscosity increases more or less exponentially with pressure

Viscosity also change with shear rate. Liquids whose viscosities are independent of the shear rate are known as "Newtonian". Liquids whose viscosities vary with shear rate are known as "non-Newtonian". The pseudoplastic fluids are characterized by linearity at extremely low and extremely high shear rates. The dilatant fluid exhibits an increase in apparent viscosity with increasing shear rate

For a range for which the effects of temperature and pressure on viscosity are found to be important, the density changes in case of lubricating liquids are small relative to the viscosity change. Extremely high pressure exists in elastohydrodynamic films and the lubricant can no longer be considered as an incompressible medium. It is therefore necessary to consider the dependence of density on pressure in such cases

The variation of density with pressure is roughly linear at low pressure but the rate of increase falls off at high pressure

The physical and chemical properties of commercial lubricants are expressed in terms of various parameters like viscosity index, flash point, pour point, saponification number, neutralization number etc

### 1.3 MODE OF OPERATION:

Various aspects of the motion of machine elements in conjunction with the lubricant give rise to different modes of operation

### 1.3.1 Hydrodynamic Lubrication:

Hydrodynamic lubrication implies a process in which two surfaces moving at some relative velocity with respect to each other are separated by a fluid film in which the pressure forces that separate the surfaces are generated by virtue of the relative motion only. The relative motion indeed generates positive pressure only if the geometrical configuration is favourable. If the relative motion tends to drag fluid from a divergent space towards a convergent space, positive pressure (pressure above atmospheric pressure) is generated. If, however, the relative motion tends to drag fluid from a convergent space to a divergent space, negative pressure (pressure below atmospheric pressure) develops

In hydrodynamic lubrication the lubricant film is generally thick so that the machine elements are prevented from coming into potential contact. However, the main drawbacks of this mode of lubrication is that at low speeds heavy loads cannot be carried and moreover there is appreciable wear because of frequent stop and start up and motion reversal

### 1.3.2 Hydrostatic Lubrication:

When the pressure to fluid film, which separates two surfaces in motion and which supports the load, is applied from an external agency (pump, accumulator, etc.), the bearing is classified as hydrostatic bearing. Hydrostatic bearings can be designed to give predetermined performance characteristics, e.g., optimization in terms of flow, pressure, load capacity, stiffness, friction and pumping power. Hydrostatic bearings have been designed to give, where necessary, extremely low coefficient of friction, and on the other hand, they have been designed to give extremely high stiffness. In large telescopes and radar tracking units hydrostatically lubricated bearings are used, where extremely heavy loads and extremely low speeds are used. In machine tools and gyroscopes also hydrostatically lubricated bearings are used as there are extremely high speeds, light loads, and gas lubricants are used. A number of studies on hydrostatic lubricated bearings have appeared [Raimondi and Boyd (1957); Ghosh and Majumdar (1978, 1980)]

### 1.3.3 Boundary Lubrication:

Boundary lubrication occurs when the intervening lubricant film between two sliding surfaces does not completely separate the surfaces. In practice, boundary lubrication covers a major portion of lubrication phenomena. Even in bearings which are designed for hydrodynamic lubrication, boundary lubrication occurs during starting, stopping, and during periods of severe operation

Sometimes due to adverse operating conditions, break down of the hydrodynamic action takes place. At low speeds or high loads, thickness of lubricant film decreases until high spots on the mating surfaces begin to rub on another and bearing surfaces no longer are completely separated by a fluid film but partial metal to metal contact takes place which increase friction and wear. This may take place in other situations as well e.g. (i) when viscosity of the lubricant is very low (ii) the system is starved off of the lubricant (iii) converging wedge is not formed. This mode of lubrication is called boundary lubrication. Boundary lubrication is not the most desirable operating mode, yet at times, it is completely unavoidable. It is encountered mainly with slow moving loads where cost of a hydrostatic bearing is prohibitive

Bearings under start or stop condition, valve without converging wedge, machine tool slide way, gears, etc. are the examples where boundary lubrication conditions exist

Mechanisms such as door hinges operate under conditions of boundary lubrication

The understanding of boundary lubrication is normally attributed to Hardy and Doubleday (1922.a, 1922.b) who found that extremely thin films adhering to surfaces were often sufficient to assist relative sliding. They concluded that under such circumstances the chemical composition of the fluid is important and they introduced the term "boundary lubrication". Boundary lubrication is at the opposite end of the lubrication spectrum from hydrodynamic lubrication. In boundary lubrication the physical and chemical properties of thin films of molecular proportions and the surfaces to which they attached determine contact behaviour. The lubricant viscosity is not an influential parameter. Because in boundary lubrication the surfaces are not separated by the lubricant, fluid film effects are negligible and there is considerable asperity contact

The frictional characteristics are determined by the properties of the moving surfaces and the lubricant film at the common interfaces. In boundary lubrication, the roughness of the surfaces plays an important part and has to be considered

### 1.3.4 Elastohydrodynamic Lubrication:

In certain situations distortion of the surfaces due to the heat generation is important which results in the changes in the film thickness which are of the same order of magnitude as the film thickness itself and this has a strong effect on the pressure development. Infact, a portion of load carrying capacity of the parallel surface bearing may be due to this phenomenon

Another source of distortion is the pressure itself which, if high enough, can distort the bearing on slider on both and in so doing change the pressure distribution

The study of this effect is called elastohydrodynamic lubrication [Dowson and Higginson (1966)]. This effect is particularly important in the lubrication of gear and roller bearings where very high pressure can be developed. In order to solve this problem which involves the interaction of elasticity and fluid flow phenomena, it is necessary to consider the film thickness in the Reynolds equation as a function of the pressure. The other equations needed are an elasticity equation that relates the displacement of the solid surfaces to the stress system [Timoshenko and Goodier (1955)] and a relationship between viscosity and pressure

The first problem to be solved seems to have been for rollers deformed by pressure generated with a fluid whose viscosity was altered by the pressure [Grubin (1949)]. Osterle and Saibel (1958) discussed the performance of the slider bearing with bearing elasticity, proving that the deflection of the bearing under high load reduces the load carrying capacity of the bearing. Ramanaiah (1968) analyzed the effect of bearing deformation in squeeze films between two long rectangular plates. It was shown that under high loads the bearing will deform producing a wedge effect in the lubricant film and the deformation decreases the load capacity of the bearing. It was assumed in the analysis [Osterle and Saibel (1958); Ramanaiah (1968)] that the deformation under the elastohydrodynamic conditions is the same as it would be under the static conditions

The effect of elastic distortion on the bearing performance have been studied among others by Carl (1964), Higginson (1965), O'Donoghue et al. (1967), Benjamin and Castelli (1970), Ibrahim and McCallion (1967) and by Oh and Huebner (1973)

Some of these works are experimental in nature, some a combination of experimental and theoretical analysis, and others purely analytical studies

## 1.3.5 Partial (Mixed) Lubrication:

Combined mode of action between fluid film and boundary lubrication is generally referred to as "mixed lubrication" or "partial lubrication". For conformal surfaces, where hydrodynamic lubrication occurs if the film gets too thin, the mode of lubrication goes directly from hydrodynamic to partial. For nonconformal surfaces, where elastohydrodynamic lubrication occurs if the film gets too thin, the mode of lubrication goes from elastohydrodynamic to partial

## 1.3.6 Turbulent Lubrication Regime:

Bearings that are operated at very high speeds, with large clearances or with lubricants having low kinematic viscosities may achieve sufficiently high values of Reynolds number in the bearing film so that some departure from laminar conditions may occur. Bearing in such a situation are said to work under turbulent regimes

## 1.3.7 Magnetohydrodynamic Lubrication (MHD):

Liquid metals like sodium and mercury as lubricants offer several advantages such as ability to with stand high temperatures over conventional lubricants. And hence the lubrication properties of liquid metals have been studied theoretically as well as experimentally. It is advantageous to use them where very high temperature and speed occur such as those in space entry vehicles. In addition, the high thermal conductivity of liquid metals means that heat generated by viscous dissipation at high speeds is readily conducted away form the source of generation, thus resulting in a tendency towards uniformity of temperature and viscosity of the lubricant film

If the liquid metals, such as mercury and sodium, could be pumped or held between the moving surfaces of the bearing, bigger loads could be supported by applying a large magnetic field

Because of the large electrical conductivity of liquid metals, the possibilities of electromagnetic pressurization from the application of an external magnetic field have been explored and studied. This electromagnetic pressurization results when a large external electromagnetic field thorough the electrically conducting lubricant is applied to induce circulating currents, which in turn, interacts with the magnetic field to create a body force which pumps the fluid between the bearing surface. The study of hydrodynamic lubrication with external electromagnetic fields is called the magnetohydrodynamic lubrication. Many interesting papers have been published discussing magnetohydrodynamic effects on the bearings. [Kuzma (1963, 1964, 1965); Kuzma, Maki and Donnelly (1964); Shukla (1963, 1964, 1965, 1970); Maki, Kuzma and Donnelly (1966); Snyder (1962); Hughes and Elco (1962.a); Elco and Hughes (1962); Hughes (1963.a, 1963.b); Dodge, Osterle and Rouleau (1965); Shukla and Prasad (1965); Dudzinsky, Young and Hughes (1967); Agrawal (1970.a, 1970.b); Anwar and Rodkiewicz (1973); Sinha and Gupta (1973)]

It may be noted that during various phases of the operation of a bearing more than one mode of operation may appear and indeed, in most practical cases this actually happens

## 1.3.8 Rarefied Gas Lubrication:

In the analysis of gas lubricated bearings, it is customary to assume that the lubricant is a continuous medium. In gas bearing applications there are occasions such as with gyroscopes and other aerospace designs that the bearings are essentially operated in the so called slip-flow regime. This can happen because of either low ambient pressure conditions or extremely thin bearing films or both

In a gas, at a considerable low pressure the length of the molecular mean free path can become comparable to the thickness of the gaseous film. The gas subjected to this condition does not behave as a continuous fluid but exhibits some characteristics of its molecular chaos. These effects may be encountered in regions having very sharp gradients of fluid properties such that these properties change sufficiently in the space of a few mean free paths, regardless of whether or not the absolute density of the gas flow is especially low

Hsing and Malanoski (1968) analyzed spiral grooved thrust bearings, by applying the proper slip flow corrections separately to the groove and the ridge region

Ramanaiah (1969) made a study of the influence of molecular mean free path on the performance of a gas lubricated thrust bearings. It was found that the effect of the slip flow is to decrease the torque on the rotor and to increase the mass flow rate a t a given feeding pressure

## 1.3.9 Porous Metal Lubrication:

Porous bearings have long been used in industry, perhaps since 1920's when the idea of using porous metals for a bearing bush of a self-lubricating bearing was first suggested probably originating from the attempts to overcome heat conductivity limitations of oil soaked wooden bearings. Analysis of porous metal lubrication was initiated by Morgan and Cameron (1957). Reynolds equation in the present case comes out to be a coupled one with the pressure in the porous region which satisfies Laplace's equation. However, incorporation of an approximation, that the thickness of the porous facing is small, results in Reynolds equation being uncoupled and takes the form of a Poisson equation. There had been a great deal of interest to study the analytical aspect of the porous metal lubrication and a lot of literature has appeared in this direction

Externally pressurized porous bearings have been found to be more useful from several applications point of view [Sneck (1968)] several investigations have been devoted towards this [Sneck and Yen (1967); Hsing (1971); Dah-Chen (1973, 1975.a, 1975.b, 1975.c); Murti (1974); Chang (1975); Majumdar (1975, 1976); Majumdar and Schmidt (1975)]. Inertia effects in porous thrust bearings have also been considered

[Hsing (1971)]

## 1.3.10 Biolubrication:

Bone terminals forming synovial joints covered by a layer of porous material called hyaline cartilage in humans and animals consist of a pair of surfaces contained within a closed cavity which contains a fluid whose mechanical properties are shear dependent. This fluid is known as synovial fluid and functions partly as a nutritional medium, exhibiting thixotropic non-Newtonian pseudoplastic behaviour [Mow (1969)]

The stress - strain behaviour of the synovial fluid may be represented by pseudoplastic power law model. [Chandra (1975)]. McCutchen (1961, 1962, 1966) suggested that the porous nature of cartilage plays an important role in the joint behaviour. He proposed that nature has ingeniously provided animals with hydrostatically lubricated joints. It was suggested that with low elastic modules and permeability of the cartilage, it is the hydrostatic pressure of the fluid within the pores that supports a large portion of the external normal traction

## 1.4 TYPES OF RELATIVE MOTION:

Bearings may be classified on the basis of the type of relative motion. Bearings based on rolling action of the two surfaces over each other are called rolling element bearings. The rolling elements might be balls, rollers or needles. Some slipping, sliding or spinning may also take place. The bearings based on sliding action or relative tangential motion of the surfaces are called plain or slider bearings. If the relative motion of the surfaces is normal to the bearing surfaces, the phenomena is called squeeze film lubrication. In such situation the bearing surfaces separated by the lubricant film approach each other and motion may be cyclic or non-cyclic. A combination of these types of relative motion may also occur. In externally pressurized bearings the motion of the lubricant is induced by external pressurization of lubricants using supply pumps etc. through a recess

## 1.5 GEOMETRY OF BEARING SURFACES:

Relative motion indeed generates positive pressures only when geometrical configuration is favourable. If the relative motion tends to drag lubricant from a divergent space towards a convergent space then only positive pressure is generated in a slider bearing. Geometry of the bearing configuration plays an important role besides the ratio of inlet to outlet film thickness in supporting the load. In slider bearings various geometrical shapes like plane inclined, composite plane, composite taper, exponential, secant shaped, parabolic curved, stepped shape, cantenoidal cycloidal and several others have been investigated [Pinkus and Sternlicht (1961); Gross et al. (1980); Bagci and Singh (1983)]. Relative transverse motion of parallel surfaces however generates a pressure which can support a transverse load. Common squeeze film bearing configurations are journal bearings consisting of cylindrical shafts, spherical and conical shapes etc. Parallel plate squeeze film bearings with circular, annular, rectangular, elliptical, triangular, sector shaped etc. have been analysed [Archibald (1956)]. Murti (1975.b) analysed curved squeeze film bearing. In externally pressurized bearings the film shapes have been considered to be uniform as well as both converging and diverging. In externally pressurized bearings also circular, rectangular and several other shapes have been investigated

## 1.6 TYPES OF LOADING:

Bearings are evaluated on the basis of their load carrying capacity at various speeds. Rolling element bearings carrying loads perpendicular to the rotational axis are called radially loaded bearings. Slider bearings carrying such loads are usually journal bearings. Thrust bearings support axial loads. In practice most bearings carry a combination of both radial and thrust loads. Load supported may be static, unidirectional one or the bearing may be dynamically loaded. In some cases transient or periodic forces or displacements are imposed on the lubricant film. Rolling element bearings generally are less sensitive than slider bearings to load variations. Fluid film bearings are a better choice if the load is dynamic

## 1.7 BEARING DESIGN CHARACTERISTICS:

Bearing design requirements are generally established by the restrictions and environmental conditions imposed by the bearing systems such as choice of lubricant, bearing material specifications, bearing life, cost, bearing alignment, positioning precision, direction and magnitude of loads, bearing ambient pressure, supply pressure, flow rate available from the system, heat flow etc. The assurance of the compatibility of the bearing and its design requires definition of both the range of imposed bearing requirements and bearing performance limitations. Following parameters that characterize the performance of a fluid film bearing are required to be designed and analyzed : (1) Lubricant flow in the bearing (2) Lubricant side leakage from the bearing (3) Pressure distribution in the film (4) Load carrying capacity (5) Centre of pressure (or attitude angle in case of journal bearing) (6) Friction force (or coefficient of friction) (7) Film stiffness (8) Squeeze film versus film thickness relationship (in case of squeeze film bearings) (9) Range for stability of the bearing both for initial velocity disturbances and initial position disturbances Usually the bearings are designed to perform optimally for a particular parameter. Typically, one or more of the following functional characteristics are required in bearing design (a) optimum load capacity (b) minimum friction (c) control of film thickness within a specified range (d) aspect of stability of the baring rotor system. (e) minimum power requirement

## 1.8 REVIEW OF RELATED LITERATURE:

During the last hundred years of analytical aspects of tribology, a lot of developments both in terms of analysis, research and developments of bearings have taken place and now the branch of tribology has gained an independent status. These developments have been documented in a number of books; few of these which have become very popular and are used as reference texts; are Pinkus and Sternlicht (1961), Fuller (1956), Tipei et al (1961), Tipei (1962), Dowson and Higginson (1966), Constantinescu (1968, 1969), Wright (1969), Moore (1972), Gross et al (1980), Szeri 36 (1980), Majumdar (1986) and Hamrock (1994). Dowson (1973) describes the early history of tribology. Few of the important research papers and survey articles include Benes (1970), Saibel and Macken (1973), Moore (1965), Sneck (1968), Wu (1978), Vinay Kumar (1980), Christensen and Tonder (1969.a), and Bagci and Singh (1983)

There are several investigations in which the generalized form of Reynolds equation is derived. Morgan and Cameron (1957) obtained the generalized Reynolds equation for porous metal bearings. Shukla (1963) derived the modified Reynolds equation for bearings working under the influence of electromagnetic fields. Dowson (1962) obtained a generalized Reynolds equation for gas bearings in which the density and viscosity variations both across and along the lubricant films were considered and Berthe and Godet (1973) derived the generalized Reynolds equation considering the bearing surface to be rough. Kulkarni and Vinay Kumar (1975) obtained the modified Reynolds equation for anisotropic porous bearings considering the tangential slip velocity at the porous wall - film interface, while Bhat (1980) generalized it further to include the electromagnetic effects, Vinay Kumar (1978) obtained the modified Reynolds equation for porous bearings in turbulent regime. Xin and Ming (1985) derived the modified Reynolds equation for porous bearings with non-uniform permeabilities considering tangential velocity slip and the effect of cavitation. Shukla and Kumar (1987) derived the modified Reynolds equation for bearings working with ferromagnetic fluids. Agrawal (1986) obtained the modified Reynolds equation for bearing working with magnetic fluids as lubricants

Reynolds (1886) considered the configuration of an elliptical plate approaching with relative velocity to another flat plate and obtained solutions for it. Underwood 37 (1945) first used the term SQUEEZE FILM for this situation. The first work on the problem appears to have been done by Stefan (1874), for a circular flat plate. Infact, the subject of squeeze film lubrication drew renewed interest in the sixties because of the possible space applications [NASA report (1969); Sneck (1968)] of squeeze film bearings such as gyrogimbal bearings. Many of its important effects have been studied theoretically as well as experimentally. Archibald (1956) presented the analysis for the squeeze film between flat surfaces. Hays (1963) considered the squeeze film phenomena between curved plates having curvature of the sine form and keeping minimum thickness as constant. Jackson (1963) included the inertia effects in his study

Moore (1965) presented an excellent review on the analyses of squeeze film bearings upto 1965. Gould (1967) investigated high pressure squeeze film for circular disks considering viscosity as function of temperature and pressure. Murti (1975.b) analysed the squeeze film between curved circular plates, curvature being expressed by exponential function with the assumption that the central film thickness remains constant. Gupta and Vora (1980) presented an analysis for the squeeze film between curved annular plates

The effect of electric and magnetic fields on the flow of electrically conducting lubricants has been studied for many years. Studies have shown that MHD bearings have several theoretical advantages over ordinary bearings. Several kinds of MHD bearings have been discussed. The most common type is the slider bearing, and two general configurations of the slider have been analysed. One configuration uses a transverse magnetic field with a tangential electric field, while the other uses a tangential magnetic field with a tangential electric field. Each of these configurations 38 has been tried with various geometrical shapes of the bearing surfaces [Snyder (1962); Elco and Hughes (1962); Hughes (1963.a, 1963.b)]. The magnetohydrodynamic composite slider bearing in the presence of transverse magnetic field has been discussed by Prakash (1967). A number of theoretical and experimental studies [Maki, Kuzma and Donnelly (1966); Snyder (1962); Hughes and Elco (1962.a, 1962.b); Hughes (1963.a, 1963.b); Shukla (1965) etc.] have been devoted to magnetohydrodynamic lubrication

Mostly, the analyses of bearing problems, the lubricant inertia is neglected in comparison to viscous forces. In most of the operations the Reynolds numbers are small enough, so that this assumption may be considered valid. However, with the continuing trend in machine design for high speeds and the use of unconventional lubricants the question of how important the effect of inertia will be at high Reynolds numbers in the laminar regime itself is of growing interest [Slezkin and Targ (1946)]. If the Reynolds number becomes sufficiently large, turbulence may develop and the governing equations may not apply even when the inertia terms are included. Several contributions including inertia effects have been made. Saibel and Macken (1974) have extensively reviewed the literature existing till that time. Inertia effects are also important because of the interest in assessing the importance of possible viscoelastic effects in lubricant behaviour and inertia both of which become important in highly unsteady conditions

Tichy and Winer (1970) studied this aspect in parallel circular squeeze film bearings and included the effect of lubricant inertia and used regular perturbation techniques

Tipei and Constantinescu (1956) for the first time used spherical polar coordinate systems for the development of the Reynolds equation for squeeze film in a spherical 39 bearings working with gas as the lubricant. Pan (1963) has analysed a problem of the hemispherical squeeze-film bearing with the spherical rotor rotating at a constant speed and also moving with a constant load. In both of the above analyses, the bearing surfaces were assumed to be smooth. Although, it has beeen realized for a long time that most sliding surfaces are rough, theoretical analysis often ignores this fact. This seems to be surprising since it is well established that the separation of the sliding surfaces in a bearing and the emplitude of the roughness are comparably in magnitude. However, bearing surfaces, particularly after having some run-in and wear, develop roughness. It has now been well established that the roughness of the surfaces significantly affects the bearing performance, especially, in bearings working in the boundary lubrication regime

Davies (1963) used a saw-tooth curve for modelling the roughness. Burton (1963) made use of a fourier type series approximation for representing a surface roughness. An example of this approach is the Michell's theory of rugolose lubrication

[Michell (1950)], where he assumes that the roughness can be represented by a lone high frequency sine wave. Tipei and Pascal (1966) extended this method by including several terms in the series approximation. However, this method is, perhaps more suitable in an analysis of the effect of waviness rather than the roughness. The basic difficulties with these methods lie in performing the actual computations. Besides the integration of Reynolds equation poses the problem

The random character of the surface roughness was recognized by several investigators who used a stochastic approach to mathematically model the roughness of the bearing surfaces [Tzeng and Saibel (1967.a, 1967.b), Christensen and Tonder 40 (1969a, 1969b, 1970.a)]. In actuality Tzeng and Saibel (1967.a) used a method of random analysis which avoided the computational difficulties in a calculation of an infinitely wide slider bearing. However, their method had only a restricted application and could not be used with surfaces of arbitrary dimensions. It appears that the intensions behind the work of Christensen and Tonder (1969.a, 1969.b), Christensen (1970) were somewhat different in that sense that they aimed towards the construction of a theory of lubrication suitable to the analysis of rough sliding surfaces in general, rather than the analysis of a particular type of rough bearing. Their approach which was based upon the concept of viewing the film thickness as a stochastic process resulted in a Reynolds type equation in the mean or expected pressure. Since this Reynolds type equation had a formed which was similar to the ordinary Reynolds equation but contained only smooth functions of film thickness, integration of Reynolds equation did not pose any problem. Christensen and Tonder (1970.a) demonstrated the power of the above theory in applying it to the analysis of a rough slider bearing of finite width. By comparing the results for different width/length ratios they concluded that the influence of surface roughness on bearing response could not be decided by the consideration of roughness alone; but that the effects of roughness were being modified by the nominal geometry as well as operational factors. In this article they concluded that roughness could increase or decrease the load capacity, influenced the frictional properties to the better or worse as compared with bearing having smooth sliding surfaces. Further the results indicated that roughness might influence the bearing characteristics considerably. Tonder (1972) theoretically analysed the transition between surface distributed waviness and random roughness. Further he claimed that the application of 41 average type relations was the only way for dealing with this kind of problem particularly in the case of two-dimensional disturbances. Christensen and Tonder (1970.b) proposed a mathematical theory of mixed lubrication based upon the stochastic theory of hydrodynamic lubrication. They developed the expressions for various bearing characteristics valid under mixed lubrication conditions. In order to illustrate some of the effects the theory was then applied to the analysis of a no side lickage, constant inclination sliding bearing. This analysis demonstrated the profound effect that transition into the mixed lubrication regime has especially on the friction dependent properties. Tzeng and Saibel (1967.a, 1967.b) used a beta probability density function for the random variable characterizing the roughness. This distribution is symmetrical in nature with zero mean and approximates the Gaussian distribution to a good degree of accuracy for certain particular cases. Christensen and Tonder (1969a, 1969b, 1970.a) further developed this approach and proposed a comprehensive general analysis both for transverse as well as longitudinal surface roughness based on a general probability density function. In squeeze film bearings when the plates are having rotatory motion also, the linear inertia may be neglected but the convective inertia due to rotatory motion may be of importance [Wu (1971); Ting (1975), Prakash and Vij (1976)]. Na (1966) obtained an inertialess solution for non-Newtonian squeeze films. Ramanaiah (1967) analysed the problem of squeeze film with power law fluid considered as lubricant and the inclusion of inertia effects. He did not include the local inertia and part of the convective inertia in the equation of motion. Elkough (1976) accounted for all the inertia terms in the equation of motion for a laminar non-Newtonian squeeze film

Ramanaiah (1966.b) analysed the problem of squeeze film between circular plates with 42 axial current induced pinch effect using power law fluid as lubricant, neglecting lubricant inertia. Gupta and Vora (1982) considered the effect of rotational inertia on the squeeze film load between porous annular curved plates

In recent years there has been a great deal of interest both from analytical and experimental point of view in the operation of bearings beyond the laminar regime

High velocity and low viscosity lead to high Reynolds numbers and departure from laminar flow. Macken and Saibel (1972) reviewed the work in this aspect

In certain situations distortion of the surface due to heat generation and heavy loading may take place and this may result in the changes in the film shape which may be of the same order of magnitude as the film thickness itself. The study of this effect is called elastohydrodynamic lubrication. This effect is particularly important in the lubrication of gear and roller bearings where high pressures develops. Osterle and Saibel (1958) discussed the performance of the slider bearing with considering the elastic deformation of the bearing. Ramanaiah (1968) analysed the elastic deformations in hydrodynamic squeeze films. Ting (1975) investigated the problem of engagement of porous plates simulating it by annular plates incorporating the effects of elastic deformation and the surface roughness of the bearings

In most of the theoretical studies of bearing lubrication, it has more or less explicitly been assumed that the bearing surfaces can be represented by smooth mathematical planes. It has, however long, been recognized [Halton (1958)] that this might be an unrealistic assumption, particularly, in bearings working with small film thickness. Several devices such as postulating a sinusoidal variation in film thickness [Burton (1963)] have been introduced in order to seek a more realistic representation of 43 engineering rubbing surfaces. However, this method is perhaps, more appropriate in an analysis of the influence of waviness rather than roughness. Tzeng and Saibel (1967.a) have introduced stochastic concepts and have succeeded in carrying through an analysis of a two dimensional inclined slider bearing with one dimensional roughness in the direction transverse to the sliding direction. However, bearing surfaces, particularly, after they have received some run-in and wear, seldom exhibit a type of roughness approximated by this model. On the contrary recent investigations demonstrate that running in and wear tend to produce a one dimensional type of roughness running in the direction of sliding or longitudinal direction [Christensen and Tonder (1969.a)]

The effect of surface roughness on the load supporting ability, friction force and oil flow has been the subject of conjecture for some time [Burton (1963); Michell (1950)], but the recent studies [Christensen and Tonder (1969.a, 1969.b, 1970.b, 1971, 1972), Citron (1962), Tzeng and Saibel (1967.b), Tipei and Pascal (1966), Tonder and Christensen (1972.a, 1972.b), Dowson and Whomes (1971), Christensen (1971) have proved its importance. A general form of the Reynolds equation has been obtained by Berthe and Godet (1973) by assuming one of the two moving contact surfaces as either rough or deformed

In 1975, Christensen et al [Christensen, Shukla and Kumar (1975)] obtained a generalized Reynolds equation applicable to rough surfaces by assuming that the film thickness function follows a stochastic process. Tonder (1977) gave a mathematical treatment of the problem of lubrication of bearing surfaces depicting two dimensional distributed uniform or isotropic roughness. Prakash and Tiwari (1982) discussed various types of roughness patterns on the bearing surfaces and solved the problem of 44 squeeze film between porous circular parallel plates. Guha (1993) investigated the effect of isotropic roughness on the performance of journal bearings

The development of magnetic fluid as lubricant has not only added to the long range of already available lubricants but also added to new characteristics of the lubricant, namely, its fluidity and magnetic property. Magnetic fluid is a multicomponent and multiphase system. It consists of fine magnetic particles coated with a surfactant and dispersed in a non-conducting and magnetically passive solvent which prevents them from aggregating. The advantage of magnetic fluid lubricant over the conventional one is that it can be retained at the desired location by appropriate application of magnetic field. In sealed systems contamination due to lubricants can be prevented if the use of magnetic fluid as lubricant is made. Under the term of magnetic fluid, a variety of liquid magnetic media, ranging from coarse dispersion to fine colloidal suspension or even molecular solution and from dielectric to very good conductors are all included. It is therefore natural that mathematical descriptions of such a variety of magnetic fluids may differ from each other. Shukla and Kumar (1987) obtained the generalized Reynolds equation for ferromagnetic lubricants and used it for slider bearing and squeeze film bearing. Zahn and Rosenweigh (1980) described the motion of magnetic fluids through porous media under the influence of obliquely applied magnetic field. This paper became the basis for many research analyses on the performance of porous bearings working with magnetic fluid as lubricant. Use of magnetic fluid as lubricant modifying the performance of the bearing has now been recognized. Agrawal (1986) analysed the problem of porous slider bearing working with the magnetic fluid as the lubricant. Bhat and Deheri (1991, 1995) also studied the problem of slider bearings lubricated with magnetic fluid. They found that slider bearings with magnetic fluid was performing better than the corresponding one lubricated with conventional lubricant

## CHAPTER – 2

## RELATED ASPECTS AND DERIVATION OF MODIFIED REYNOLDS EQUATION

In this chapter, we propose to develop the generalized Reynolds equation

### 2.1 INTRODUCTION:

The basic equation employed in the analysis of fluid film lubrication is called the Reynolds equation which was first presented by Reynolds in 1886 which was formed by combining the equation of motion and the equation of continuity. In deriving this equation Reynolds neglected fluid inertia and gravitational effects in relation to viscous action restricting his analysis to a thin film of isoviscous, incompressible fluid. The adequacy of these assumptions was demonstrated by the satisfactory explanation of the performance of a number of non-porous bearings. However, the application of hydrodynamic theory within the assumptions made by Reynolds is valid only over a much narrower field than is generally supposed [Halton (1958)]; in particular, surfaceroughness, high speed variable viscosity, thermal effects etc. emphasize the need to generalize the Reynolds equation accordingly. Moreover, the increased severity of bearing operating conditions, the greater use of gas bearings, the porous bearings and several limitations pertaining to lubricant properties etc. have also necessitated the generalization of Reynolds equation to account for the various effects. However, to improve upon the bearing performance which otherwise suffers on many counts, several attempts have been made incorporating the various factors such as surface roughness etc. arising out of recent technological advancements

For the sake of completeness we discuss, below, the gradual development of the Reynolds equation incorporating the various effects that come to be accounted for and gradual relaxing of the various assumptions made in the theoretical investigations to approach nearer the more realistic situations

## 2.1.1 Inertia and Turbulent Effects in Lubrication:

In most hydrodynamic bearings the lubricant flow is laminar and as such is governed by the Navier-Stokes equations which relate the pressure and viscous forces acting on the lubricant to the lubricant inertia. The importance of the inertia terms relative to the viscous terms in this equation can be characterized by a parameter known as Reynolds number which increases as the inertia effect increases. In the operation of most bearings the Reynolds numbers are small enough, so that the inertia effect can be ignored safely, reducing the governing relationship to the familiar Reynolds equation

However, with the continuing trend in machine design for which higher speeds as well as the use of unconventional lubricants such as water or liquid metals, the question of how important inertia will be at high Reynolds numbers in the laminar regime itself is of growing interest [Slezkin and Targ (1946); Kahlert (1948)]. If the Reynolds number becomes sufficiently high, turbulence may develop and the previous governing equation will no longer apply even with the inertia terms included. Several contributions towards including the inertia effects and their re-examinations have been made [Brand (1955); Osterle and Saibel (1955.a); Osterle, Chou and Saibel (1957); Milne (1959); Snyder (1963)]. Inertia effects are also important because of the current interest in assessing the importance of possible visco-elastic effects in lubricant behaviour. Both visco-elasticity and inertial effects are likely to become important in highly unsteady conditions

Therefore, it is necessary to have a through understanding of inertia effects in order to adequately isolate them as not to inadvertently attribute them to visco-elasticity

The relative importance of the various fluid inertia terms in the inertia force has been based on the order of magnitude analysis and consequently with varying degree of approximations for simplifying the analysis; various studies have been made [Tichy and Winer (1970); Pinkus and Sternlicht (1961); Saibel and Macken (1974); Jones and Wilson (1975)]. The retention of inertial terms in the Navier-Stokes equations gives rise to non-linearity, resulting in the analysis becoming quite complicated. However, the method of averaged inertia and the method of iteration have been used to account for inertia terms [Slezkin and Targ (1946); Kahlert (1948); Agrawal (1969.a, 1969.b, 1970.c, 1970.d)]. Tichy and Winer (1970) used the method of regular perturbation taking Reynolds number as the perturbation parameter, while Rodkiewicz and Anwar (1971) used the series-expansion method

### 2.1.2 Thermal Effects in Lubrication:

Research into the thermodynamics of fluid film bearings got an impetus by the experimental work of Fogg (1946) who observed that a parallel surface thrust bearing can support a load and gave an explanation for this by introducing the concept of thermal wedge i.e. the expansion of fluid due to heating. Later, Cope (1949) modified the classical Reynolds equation by introducing viscosity and density variation along the fluid film and obtained a corresponding form of the energy equation for determining temperature in the film. He coupled the energy balance equation in the fluid film with the momentum and continuity equations to obtain the temperature and the pressure distribution. This form of the equation differed from those of Christopherson (1941), and Cameron and Wood (1959) and this discrepancy was due to the neglect of the work done by the pressure forces in the energy equation [Charnes, Osterle and Saibel (1952)]

The effect of viscosity variation due to pressure and temperature on the characteristics of the slider bearing have been studied by Charnes, Osterle and Saibel 50 (1953.a, 1953.b, 1955); Osterle and Saibel (1955.b). The variation of viscosity across the film thickness among others has been studied by Zienkiewicz (1957); Cameron (1960), who concluded that the temperature gradients and viscosity variation across the film should not be ignored. Dowson (1962) generalized the Reynolds equation by taking into consideration variation of fluid characteristics across the film thickness. Dowson and Hudson (1963.a) used the generalized Reynolds equation, allowing for the heat transfer to the bearing solids to determine the fluid film thermal boundaries. They also [Dowson and Hudson (1963.b)] studied the case of parallel surface bearing by taking into account the heat transfer to the bearing surfaces and found that their investigation completely reversed the earlier predictions of the thermal and viscosity wedge effects

The effect of thermal distortions of the bearing solids was considered by Hahn and Kettleborough (1967, 1968, 1969) in an analysis of the one dimensional slider bearing

They concluded that while thermal distortions are responsible for the load carrying capacity exhibited by parallel surface sliders, they have little effect on the performance of inclined sliders when compared to leakage effects

Gould (1967) discussed the thermohydrodynamic performance of fluid film between two surface approaching each other at a constant velocity, the effect of prescribed thermal gradients was also studied for journal bearings [Hagg (1944); Hughes and Osterle (1958); Motosh (1964); Tipei and Nica (1967)]. McCallion et. al

(1970) presented a thermohydrodynamic analysis of a finite journal bearing. The analysis indicated that the bearing load carrying capacity is insensitive to heat transfer in the solids. It also showed that the thermohydrodynamic load is not necessarily bounded by either the adiabatic or the isothermal solutions

### 2.1.3 Non-Newtonian Lubricants:

In the development of better lubrications required to meet the needs of advancing scientific technology, it was found that by adding polymer additives to the base lubricant, the viscosity of the fluid is increased considerably and is relatively temperature independent. This increase in viscosity brings about an increase in load carrying capacity. It was observed that the viscosity of the modified lubricant is, however, no longer constant, but decreases as the rate of strain increases. This particular phenomenon is known as pseudoplastic behaviour. It can be quantitatively explained by assuming that the initially random molecules of the polymer additives in the lubricant become realigned in the direction of motion when placed in a shear field. For example, in order to have engine performance fairly uniform over a wide range of temperature and pressure, engine oils have been treated with additives

Steidler and Horowitz (1960) analyzed mathematically the effect of non- Newtonian lubrication on the slider bearing with side leakage. The corresponding experimental analysis was carried out by Dubois et al. (1960). Using a perturbation technique and taking n = 3, Saibel et al. (1962) gave a solution of the slider bearing with side leakage and found that the load carrying capacity is reduced by about ten percent

Tipei and Rohde (1974) gave a new theological model for lubricants containing additives having long molecules. The viscosity of the fluid depends on the angle between viscous forces and the velocity vector at each point in the lubricating film

They discussed the finite slider bearing lubricated with such a lubricant. Kodnir et al

(1975) obtained an approximate solution of the stationary, isothermal elastohydrodynamic problem for a Ree-Eyring fluid model, also the solution's algorithm is described for a non-Newtonian fluid of an arbitrary model. A similar problem has been solved by Chow and Saibel (1971), Bell (1972) studied the effect of lubrication on rolling surfaces when the lubricant used was a Ree-Eyring fluid

### 2.1.4 Surface Roughness Effects in Lubrication:

Halton (1958) recognized that, in bearings working with small film thickness it was unrealistic to assume that the bearing surfaces can be represented by smooth mathematical planes. A more realistic representation of engineering rubbing surfaces was presented by Burton (1963) by critically examining the Reynolds equation

However, his method was inappropriate for the general type of roughness. Tzeng and Saibel (1967.a) studied the effect of surface roughness by introducing stochastic concepts related to roughness; while deriving the Reynolds equation. This method also fell short of approximating the roughness. It was the contribution of Christensen and Tonder (1969.a, 1969.b, 1970.a, 1970.b) who proposed a statistical analysis in order to obtain a modified Reynolds equation and analyzed the surface roughness effect. They concluded that the bearing performance could decrease or increase depending on the type of roughness. In 1975 Christensen et al. (1975) obtained a generalized Reynolds equation applicable to rough surfaces by assuming that the film thickness function is a stochastic process. Christensen and Tonder's approach was employed by Gupta and Deheri (1996) in order to study the effect of transverse surface roughness on the behaviour of squeeze film in a spherical bearing. In this paper it was concluded that the bearing suffrs on account of transverse roughness. Subsequently, Anhdaria, Gupta and Deheri (1997) extended the method of Christensen and Tonder by incorporating the measure of symmetry and mean of roughness besides the standar deviation, while deriving the modified Reynolds equation, and analyze the problem of the effect of this generalized roughness on the performance of slider bearings. It has been established in these investigations that the bearing performance suffers mostly because of transverse roughness, while the standard deviation of roughness may enhance the performance of a longitudinally rough slider bearing

Chow and Cheng (1976) extended Christensen's approach (1971) to determine the surface roughness influence on the inlet film thickness of EHD contacts. Kodnir and Zhilnikov (1976) obtained the solutions of steady state elastohydrodynamic problems with roughness. Tonder (1977) gave a mathematical treatment of the problem of lubrication of bearing surfaces depicting two dimensional distributed uniform or isotropic roughness. Shukla et al. (1974) derived a generalized form of Reynolds equation and integrated form of energy equation to study the thermal effects with surface roughness. Burton (1973) has considered roughness effects in turbulent lubrication

### 2.2 MATHEMATICAL MODELLING OF A BEARING SYSTEM:

The mathematical modelling of the bearing system is closely linked to the research developments in the field of fluid dynamics of real fluids which started in nineteenth century. Hydrodynamic film lubrication was effectively used before it was scientifically understood. The process of lubrication is basically a part of an overall phenomena of hydrodynamics whose scientific analysis was initiated during nineteenth century. Adams (1853) first attempted, developed and patented several rather good designs for railway axle bearings in 1847. The understanding of hydrodynamic lubrication began with the classical experiments of Tower (1883, 1884, 1885) in connection with the investigation of friction of the railway partial journal bearing when he measured the lubricant pressure in the bearing. Petrov (1883) in his separate independent studies reached the same conclusion from friction measurements. This work was closely followed by Reynolds (1886). He applied hydrodynamic laws to the bearing problem and was able to explain Tower's results satisfactorily. He derived and employed an equation for the analysis of fluid film lubrication which has by now become a basic governing equation and is named after him as Reynolds equation. He has combined Navier-stokes equations with continuity equation to generate a second order differential equation for lubricant pressure. This equation is derived under certain assumptions, such as neglect of inertia and gravitational effects in comparison to viscous action, lubricating film to be a thin one of isoviscous incompressible fluid etc

This equation can be deduced from first principles also provided same set of assumptions is made. Although, the adequacy of these assumptions was demonstrated by the satisfactory explanation of the performance of a number of bearings, however, subsequently it was realized that the Reynolds equation is valid only over a much narrower field than is generally supposed. The so called conventional Reynolds equation contains viscosity, density and film thickness as parameters. These parameters both determine and depend on the temperature and the pressure fields and on the elastic behaviour of the bearing surfaces. Besides these, sometimes surface roughness, porosity and other increased severity of bearing operating conditions etc. may demand the need to generalize Reynolds equation accordingly to account for these effects. Likewise, consistent with these effects and the requirement of the particular bearing problems, it may become necessary to relax few of the assumptions used for derivation of the Reynolds equation. Thus, study of hydrodynamic lubrication is from a mathematical point of view is infact, the study of a particular form of Navier-Stokes equations compatible with the system. Since the Reynolds time, researches in the field of lubrication have made much progress and with the rapid advancement of machines, manufacturing process and materials in which lubrication plays an important role, the study of lubrication has gained considerable importance and has become, from analytical point of view, an independent branch of fluid mechanics. From practical point of view it remains a part of TRIBOLOGY

Mathematical modelling of a bearing system consists of various conservation laws of fluid dynamics such as conservation of mass, momentum, energy and equation describing various aspects characterizing the bearing problem such as constitutive equation of lubricant, viscosity dependence on pressure - temperature, equation of state, elastic deformations, surface roughness etc.

[…]

### 2.4 BASIC ASSUMPTIONS OF HYDRODYNAMIC LUBRICATION:

The general mathematical model described above is highly non-linear in character besides being a coupled one. Thus, the severe complexity of the mathematical system describing the general problem of lubrication, theoretically, does not lend it at all straight to analytical study. A number of simplifications resulting from the physical considerations compatible with the system are required to be made before attempting to proceed to solve the system. Simplifications may be of great value if their limitations are clearly specified. It is of prime importance that all assumptions or simplifications be justified and that the limitations imposed thereby be understood in interpreting the results. Likewise, in certain situations certain idealizations may be required to be made and consequently the limits of their applicability must be recognized. Order of magnitude analysis may be attempted to estimate the relative effects of various terms in the equations and hence to simplify it. Assumptions that are to be made and the simplifications resulting therefrom would depend upon the nature of the problem and the aspect of the problem to be studied

For the analysis that follows to derive the modified Reynolds equation following assumptions are usually made : (1) The lubricant is considered to be incompressible, non-conducting and nonmagnetic with constant density and viscosity, unless and otherwise stated. Most lubricating fluids satisfy this condition

(2) Flow of the lubricant is laminar, unless and otherwise stated. A moderate velocity combined with a high kinematic viscosity gives rise to a low Reynolds number, at which flow essentially remains laminar

(3) Body forces are neglected, i.e. there are no external fields of force acting on the fluid. While magnetic and electrical forces are not present in the flow of nonconducting lubricants, forces due to gravitational attraction are always present

However, these forces are small compared to the viscous force involved, they are usually neglected in lubrication mechanics without causing any significant error

(4) Flow is considered steady, unless and otherwise stated, i.e. velocities and fluid properties do not vary with time. Temporal acceleration due to velocity fluctuations are small enough in comparison with lubricant inertia, hence may usually be ignored

(5) Boundary layer is assumed to be fully developed throughout the lubricating region so that entrance effects at the leading edge and the film discontinuity at the trailing edge from which vortices may be shed, are neglected

(6) A fundamental assumption of hydrodynamic lubrication is that the thickness of the fluid film is considered very small in comparison with the dimensions of the bearings. As a consequence of this assumption : (a) the curvature of the film may be neglected, so that bearing surfaces may be considered locally straight in direction

(b) Fluid inertia may be neglected when compared with viscous forces

(c) Since lubricant velocity along the transverse direction to the film is small, variation of pressure may also be neglected in this direction

(d) Velocity gradients across the film predominate as compared to those in the plane of the film

(7) The fluid behaves as a continuum which implies that pressure are high enough so that the mean free path of the molecule of the fluid are much smaller than the effective pore diameter or any other dimension. No slip boundary condition is applicable at the bearing surfaces

(8) Lubricant film is assumed to be isoviscous

(9) Temperature changes of the lubricant are neglected

(10) The bearing surfaces are assumed to be perfectly rigid so that elastic deformation of the bearing surfaces may be neglected

(11) In case of bearings working with magnetic fluids, the lubricant is assumed to be free of charged particles

(12) When bearings work under the influence of electromagnetic fields, it is assumed that the forces due to induction are small enough to be neglected

### 2.5 MODIFIED REYNOLDS EQUATION:

The differential equation which is developed by making use of the assumptions of hydrodynamic lubrication in equations of motion and continuity equation and combining them into a single equation governing lubricant pressure is called Reynolds equation. The Reynolds equation when derived for more general situations like porous bearings or hydromagnetic bearings or bearings working with non-Newtonian or magnetic lubricant, etc. is called generalized Reynolds equation or modified Reynolds equation. This equation is the basic governing differential equation for the problems of hydrodynamic lubrication

The differential equation originally derived by Reynolds is restricted to incompressible fluids. This, however is an unnecessary restriction, for the equation can be formulated broadly enough to include effects of compressibility and dynamic loading. We have called this the generalized Reynolds equation

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- Quote paper
- Paresh Andharia (Author), 2000, Numerical modelling of certain problems of lubrication. The effect of transverse and longitudinal surface roughness on the performance of the bearing system, Munich, GRIN Verlag, https://www.grin.com/document/323514

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