DEVELOPMENT OF REYNOLDS EQUATION

Transcription

1 CHAPTER DEVELOPMENT OF REYNOLDS EQUATION.1 Introduction.1.1 Inertia and Turbulent Effects in Lubrication.1. Termal effects in Lubrication.1.3 Non-Newtonian Lubricants. Matematical Modelling of a Bearing System..1 Equation of State.. Constitutive Equation of Lubricant..3 Continuity Equation..4 Te Equations of Motion..5 Energy Equation..6 Elastic Considerations.3 Boundary conditions.4 Basic Assumptions of Hydrodynamic Lubrication.5 Modified Reynolds Equation.6 Magnetic Fluid as a Lubricant.7 Surface Rougness 9

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3 .1 INTRODUCTION Reynolds (1886) presented te basic equation governing te analysis of fluid film lubrication. Tis equation in fact, arose from te combination of equation of motion and equation of continuity. Wile deriving tis equation Reynolds neglected fluid inertia and gravitational effects in relation to viscous action restricting is analysis to a tin film of isoviscous, incompressible fluid. However, te application of ydrodynamic teory witin te assumptions made by Reynolds is valid only over a muc narrower field tan is generally supposed [Halton (1958)]; in particular, surface-rougness, ig speed variable viscosity, termal effects etc. empasie te need to generalie te Reynolds equation accordingly. Moreover, te increased severity of bearing operating conditions, te porous bearing and several limitations pertaining to lubricant properties etc. ave also necessitated te generaliation of Reynolds equation to account for te various effects. For te sake of completeness we discuss below, te gradual development of te Reynolds equation incorporating te various effects tat come to be accounted for and gradual relaxing of te various assumptions made in te teoretical investigations to approac nearer to te more realistic situations..1.1 INERTIA AND TURBULENT EFFECTS IN LUBRICATION In most ydrodynamic bearings te lubricant flow is laminar and as suc is governed by te Navier-Stokes equations wic relate te pressure 30

4 and viscous forces acting on te lubricant to te inertia. Te importance of te inertia terms relative to te viscous terms in tis equation can be caracteried by a parameter known as Reynolds number wic increases as te inertia effect increases. In te operation of most bearings te Reynolds numbers are small enoug, so tat te inertia effect can be ignored safely, reducing te governing relationsip to te familiar Reynolds equation. However, wit te continuing trends in macine design for wic iger speeds as well as te use of unconventional lubricants suc as water of liquid metals, te question of ow important inertia will be at ig Reynolds numbers in te laminar regime itself is of growing interest [Slekin and Targ (1946); Kalert (1948)]. If te Reynolds number becomes sufficiently ig, turbulence may develop and te previous governing equation will no longer apply even wit te inertia terms included. Several contributions towards including te inertia effects and teir re-examinations ave been made [Brand (1955); Osterle and Saibel (1955.a); Osterle, Cou and Saibel (1957); Milne (1959); Snyder (1963)]. Inertia effects are also important because of te current interest in assessing te importance of possible visco-elastic effects in lubricant beaviour. Bot visco-elasticity and inertial effects are likely to become important in igly unsteady conditions. Terefore, it is necessary to ave a toroug understanding of inertia effects in order to adequately isolate tem as not to inadvertently attribute tem to viscoelasticity. Te relative importance of te various fluid inertia terms in te inertia force as been based on te order of magnitude analysis and 31

5 consequently wit varying degree of approximations for simplifying te analysis; various studies ave been made [Ticy and Winer (1970); Pinkus and Sternlict (1961); Saibel and Macken (1974); Jones and Wilson (1975)]. Te retention of inertial terms in te Navier-Stokes equations gives rise to non-linearity, resulting in te analysis becoming quite complicated. However, te metod of averaged inertia and te metod of iteration ave been used to account for inertia terms [Slekin and Targ (1946); Kalert (1948); Agrawal (1969.a, 1969.b, 1970.c, 1970.d)]. Ticy and Winer (1970) used te metod of regular perturbation taking Reynolds number as te perturbation parameter, wile Rodkiewic and Anwar (1971) used a series expansion metod..1. THERMAL EFFECTS IN LUBRICATION Researc into te termodynamics of fluid film bearing got an impetus by te experimental work of Fogg (1946) wo observed tat a parallel surface trust bearing can support a load and gave an explanation for tis by introducing te concept of termal wedge i.e. te expansion of fluid due to eating. Later, Cope (1949) modified te classical Reynolds equation by introducing viscosity and density variation along te fluid film and obtained te corresponding form of te energy equation for determining temperature in te film. He coupled te energy balance equation in te fluid film wit te momentum and continuity equations to obtain te temperature and te pressure distribution. Te effect of viscosity variation due to pressure and temperature on te caracteristics of te slider bearing ave been studied by Carnes, 3

6 Osterle and Saibel (1953.a, 1953.b, 1955); Osterle and Saibel (1955.b). Te variation of viscosity across te film tickness among oters as been studied by Cameron (1960), wo concluded tat te temperature gradients and viscosity variation across te film sould not be ignored. Dowson (196) generalied te Reynolds equation by taking into consideration variation of fluid caracteristics across te film tickness. Dowson and Hudson (1963.b) studied te case of parallel surface by taking into account te eat transfer to te bearing surfaces and found tat teir investigation completely reversed te earlier predictions of te termal and viscosity wedge effects. Gould (1967) discussed te termoydrodynamic performance of fluid between two surface approacing eac oter at a constant velocity. McCallion et al. (1970) presented a termoydrodynamic analysis of a finite journal bearing. Te analysis indicated tat te bearing load carrying capacity is insensitive to eat transfer in te solids. It also sowed tat te termoydrodynamic load is not necessarily bounded by eiter te adiabatic or te isotermal solutions..1.3 NON-NEWTONIAN LUBRICANTS In te development of better lubricants required to meet te needs of advancing scientific tecnology, it was found tat by adding polymer additives to te base lubricant, te viscosity of te fluid is increased considerably and is relatively temperature independent. Tis increase in viscosity brings about an increase in load carrying capacity. It was 33

7 observed tat te viscosity of te modified lubricant is, owever, no longer constant, but decreases as te rate of strain increases. Steidler and Horowit (1960) analyed matematically te effect of non-newtonian lubrication on te slider bearing wit side leakage. Te corresponding experimental analysis was carried out by Dubois et al. (1960). Using a perturbation tecnique and taking n=3, Saibel (196) gave a solution of te slider bearing wit side leakage and found tat te load carrying capacity is reduced by about ten percent.. MATHEMATICAL MODELLING OF A BEARING SYSTEM Te matematical modelling of te bearing system is closely linked to te researc developments in te field of fluid dynamics of real fluids wic started in nineteent century. Hydrodynamic film lubrication was effectively used before it was scientifically understood. Te process of lubrication is basically a part of overall penomena of ydrodynamics wose scientific analysis was initiated during nineteent century. Adams (1853) first attempted, developed and patented several rater good designs for railway axle bearing in Te understanding of ydrodynamic lubrication began wit te classical experiments of Tower (1883, 1884, 1885) in connection wit te investigation of friction of te railway partial journal bearing wen e measured te lubricant pressure in te bearing. Reynolds (1886) derived and employed an equation for te analysis of fluid film lubrication wic as by now become a basic governing equation and is named after im as Reynolds equation. He as combined Navier-Stokes equations wit continuity equation to generate a 34

8 second order differential equation for lubricant pressure. Tis equation is derived under certain assumptions, suc as neglect of inertia and gravitational effects in comparison to viscous action, lubricant film to be a tin one of isoviscous incompressible fluid etc. Te conventional Reynolds equation contains viscosity, density and film tickness as parameters. Tese parameters bot determine and depend on te temperature and te pressure fields and on te elastic beaviour of te bearing surfaces. Besides tese, sometimes surface rougness, porosity and oter increased severity of bearing operating conditions etc. may demand te need to generalie Reynolds equation accordingly to account for tese effects. Likewise, consistent wit tese effects and te requirement of te particular bearing problems, it may become necessary to relax few of te assumptions used for derivation of te Reynolds equation. Tus, study of ydrodynamic lubrication is from a matematical point of view is in fact, te study of a particular form of Navier-Stokes equations compatible wit te system. Since te Reynolds time, researces in te field of lubrication ave made muc progress and wit te rapid advancement of macines, manufacturing process and materials in wic lubrication plays an important role, te study of lubrication as gained considerable importance and as become, from analytical point of view, an independent branc of fluid mecanics. From practical point of view it remains a part of TRIBOLOGY. Matematical modelling of a bearing system consists of various conservation laws of fluid dynamics suc as conservation of mass, momentum, energy and equation describing various aspects caracteriing 35

9 te bearing problem suc as constitutive equation of lubricant, viscosity dependence on pressure temperature, equation of state, elastic deformations, surface rougness etc...1 EQUATION OF STATE Penomenological consideration required specification of te state of fluid wic is given by an equation wic is called equation of state. For an incompressible fluid it is given by =constant (..1) wile for a perfect gas for isotermal variations in pressure it is given by Boyle-Mariotte law as P R T (..) were Ris te universal gas constant. For constant compressibility fluids under isotermal conditions, equation of state is 0 P 0 exp C P (..3) were 0 is te value of at te reference atmosperic pressure Pand 0 C is te compressibility. Tis particular equation of state applies rater well to most liquids. is For ideal gas flow under adiabatic condition, te equation of state 36

10 0 P P 0 (..4) were C P CV is te ratio of specific eats at constant pressure and constant volume of te fluid. as For many termodynamic processes te equation (..4) is written n P 0 P (..5) 0 were 1 n, tis equation is called polytropic law... CONSTITUTIVE EQUATION OF LUBRICANT Te matematical equation relating te viscous contribution to te stress tensor wit te rate of deformation tensor is called constitutive equation applicable to te description of reological beaviour of te lubricant. Te constitutive equations are of tree types namely integral type, rate type, and differential type. Many lubricating fluids are generally Newtonian and in suc case, searing stress is directly proportional to rate of strain tensor, constant of proportionality being te dynamic viscosity of te lubricant being Newtonian in caracter greatly simplifies te matematical analysis. Te lubricants wic exibit a 37

11 relationsip oter tan tat exists for a Newtonian lubricant are generally called non-newtonian lubricant. In case of ideal plastic like grease, some initial stress must be imposed upon te lubricant before te flow begins. Minimum stress necessary to cause te flow is called yield stress. Minimum stress necessary to cause te flow is called yield stress. A real plastic beaves in a non-newtonian manner up to certain searing stress and ten starts to beave as if it is Newtonian. A number of fluids ave been classified on te basis of teir constitutive equation and given various names e.g. Maxwell fluid, second order fluid, Walter s fluid, Oldroyd fluid, etc. A general class of non-newtonian fluids for wic stress tensor is expressed as directly proportional to some power n of deformation tensor is called power law fluid. If n 1, te fluid is called dilatant and for n 1, it is termed as psudo-plastic. Tere are certain lubricants called tixotropic and in te case of viscosity decreasing wit time tey are called reopectic. Lai, Kuei and Mow (1978) ave given a list of constitutive equations for te reological beaviour of te synovial fluid wic acts as lubricant in synovial joints of uman body. Significant lubricating fluid properties are viscosity, density, specific eat and termal conductivity. Among tese fluid properties, viscosity plays a more prominent role. Viscosity varies wit temperature as well as pressure and tis variation is important in lubrication mecanics. 38

12 Te viscosity of eavily loaded lubricating film is generally treated as a function of bot pressure and temperature. According to Barus (1893) viscosity may be approximated for limited ranges by a P P bt (..6) 0 exp 0 T0 were a and b are called pressure and temperature viscosity coefficient and te subscript 0 refers to atmosperic conditions. Over reasonably large, ranges of temperature and pressure, te linear relation. P P bt (..7) 0 exp 1 a 0 T0 is useful...3 CONTINUITY EQUATION All fluid flow problems satisfy te basic law of conservation of mass, besides laws of conservation of momentum and energy. Te equation expressing law of conservation of mass is called continuity equation. It expresses te condition tat for any fixed volume of sourcesink free region, te mass of entering fluid must equal te mass of fluid leaving plus accumulated mass. If te fluid is compressible, te continuity equation is. t q 0 (..8) 39

13 were q is te velocity vector of te flowing fluid and is te density. If te flow is steady 0 t and ence te continuity equation becomes q 0. (..9) Te equation of continuity for omogeneous, incompressible fluid takes te form. q 0 (..10) A comparison of equations (..8) and..10) sows tat te density of te fluid does not appear in te continuity equation for incompressible fluids wereas it does appear in te corresponding equation for compressible fluids. Tus te continuity equation for incompressible fluids is a purely kinematical equation wereas for compressible fluids it is a dynamical one...4 THE EQUATIONS OF MOTION Principle of conservation of momentum wen applied to fluid contained in a control volume states tat forces acting on te fluid in te control volume equal te rate of outflow of momentum from te control volume troug te closed surface enclosing it. Te matematical equation expressing tis condition for Newtonian, isoviscous, laminar, continuum and compressible fluid flow for wic body forces suc as gravitational forces or electromagnetic forces etc. Are considered negligible is: 40

14 q q. q p.q q (..11) t were is called te coefficient of sear viscosity of te fluid and is called coefficient of bulk viscosity. It is often assumed tat tey are related by 3 0. Te equation (..11), were first obtained by Navier in 181 and later independently by Stokes in Hence tese are known as Navier-Stokes equations. Te first term on te left and side of equation (..11) is temporal acceleration term wile te second is convective inertia term. Te first term on rigt and side is due to pressure and te oter terms are viscous forces. If, owever, te fluid is incompressible, as is te case wit most liquid lubricants, ten. q 0 and equation (..11) simplifies to q q. q p q (..1) t Wen a large external electromagnetic field troug te electrically conducting lubricant is applied it gives rise to induced circulating currents, wic in turn interacts wit te magnetic field and creates a body force called Lorent force. Tis extra electromagnetic pressuriation pumps te fluid between te bearing surfaces. In suc a case Navier- Stokes equations for an incompressible isoviscous liquid get modified as q q. q p q J B (..13) t 41

15 were J is te electric current density and B is te magnetic induction vector. In tis case, Maxwell s equations and Om s law sould also to be taken into account. Tese are: B 0 J,. B 0 J E q B (..14) E 0,. E 0 Were E is te electric field intensity vector, is te electrical conductivity and 0 is te magnetic permeability of te lubricant...5 ENERGY EQUATION Tis equation formulates te conservation of energy principle for a fluid element in wic tere is no eat source or sink. Te equation is: C VT g t q. C T P.q. K. T V (..15) were C V = Specific eat at constant volume per unit weigt, T = Absolute temperature, K = Coefficient of termal conductivity, = Viscous dissipation function. 4

16 Te above equation states tat witin an element volume te rate of cange of internal energy plus compression work must be balanced by te energy conducted by te fluid and dissipated by friction. Te first term on te left and side denotes adiabatic compression work and te first term on rigt and side is te conductive eat transfer. For incompressible flow of lubricant adiabatic compression work energy becomes ero. Te energy equation does not apply wen te film lubricated bearing during operation is isotermal...6 ELASTIC CONSIDERATIONS In some cases, termal stresses or ig loading of te bearing surfaces may distort te film sape and consequently may affect te pressure distribution. Te study of tis aspect of lubrication is called elastoydrodynamic lubrication. Tis effect is particularly important in te lubrication of gear and roller bearings were very ig pressure can be developed. In order to matematically model suc a system, wic involves te interaction of elastic and fluid penomena, additional equation to account for elastic deformations is needed. Tis equation is called elasticity equation; it relates te displacements of te solid surface to te stress system..3 BOUNDARY CONDITIONS Te matematical description of lubrication problem formulated above is in te form of differential equations wic is required to be solved and ence until a set of boundary conditions compatible wit te 43

17 pysical system is not prescribed, te matematical description of te model would not be complete. It te film conditions are steady, te momentum equations may be combined wit te continuity equation to give equation governing te film pressure - tis equation is called REYNOLDS equation wic is a single differential equation relating pressure, density, surface velocities and film tickness. However, before tese equations can be integrated, it is necessary to establis te boundary conditions. Te combination of momentum and continuity equations allows te distributed film velocity to be eliminated and replaced by te film surface velocities. Unless slip is present tese film surface velocities are identical to te velocities of te adjacent bearing surfaces tis condition is called no-slip condition. For te values of Knudsen number, K n (= l were lis te mean free molecular pat and is film tickness) less tan 0.01, flow may be treated as continuum and no-slip conditions may applied. Wen 0.01 Kn 15slip flow becomes significant and for K n 15 fully developed molecular flow results. As regards te boundary conditions for te film pressure governed by Reynolds equation te entrance and exit effects of a self acting slider bearing may usually be ignored. Pressure is terefore taken to be ambient along te boundary. Since te ydrodynamic pressures generated in te film are very large compared to te ambient pressures, te pressure conditions are usually assumed to be ero. Te pressure condition at te source inlet of an externally pressuried film is given by te value of te supply pressure. 44

18 However, from of te boundary conditions for a particular problem depends upon te peculiarities of te particular situation..4 BASIC ASSUMPTIONS OF HYDRODYNAMIC LUBRICATION Te general matematical model described above is igly nonlinear in caracter besides being a coupled one. Tus, te severe complexity of te matematical system describing te general problem of lubrication, teoretically, does not lend it at all straigt to analytical study. A number of simplifications resulting from te pysical considerations compatible wit te system are required to be made before attempting to proceed to solve te system. Simplifications may be of great value it teir limitations are clearly specified. It is of prime importance tat all assumptions or simplifications be justified and tat te limitations imposed tereby be understood in interpreting te results. Likewise, in certain situations certain idealiations may be required to be made and consequently te limits of teir applicability must be recognied. Order of magnitude analysis may be attempted to estimate te relative effects of various terms in te equations and ence to simplify it. Assumptions tat are to be made and te simplifications resulting tere from would depend upon te nature of te problem and te aspect of te problem to be studied. For te analysis tat follows to derive te modified Reynolds equation, following assumptions are usually made: 45

19 1 Te lubricant is considered to be incompressible, nonconducting and non-magnetic wit constant density and viscosity, unless oterwise stated. Most lubricating fluids satisfy tis condition. Flow of te lubricant is laminar, unless oterwise stated. A moderate velocity combined wit a ig kinematic viscosity gives rise to a low Reynolds number, at wic flow essentially remains laminar. 3 Body forces are neglected, i.e. tere are no external fields of force acting on te fluid. Wile magnetic and electrical forces are not present in te flow of non-conducting lubricants, forces due to gravitational attraction are always present. However, tese forces are small enoug as compared to te viscous force involved. Tus, tey are usually neglected in lubrication mecanics witout causing any significant error. 4 Flow is considered steady, unless oterwise stated, i.e. velocities and fluid properties do not vary wit time. Temporal acceleration due to velocity fluctuations are small enoug in comparison wit lubricant inertia, ence may usually be ignored. 5 Boundary layer is assumed to be fully developed trougout te lubricating region so tat entrance effects at te leading 46

20 edge and te film discontinuity at te trailing edge from wic vortices may be sed, are neglected. 6 A fundamental assumption of ydrodynamic lubrication is tat te tickness of te fluid is considered very small in comparison wit te dimensions of te bearings. As a consequences of tis assumptions: Te curvature of te film may be neglected, so tat bearing surfaces may be considered locally straigt in direction. Fluid inertia may be neglected wen compared wit viscous forces. Since lubricant velocity along te transverse direction to te film is small, variation of pressure may also be neglected in tis direction. Velocity gradients across te film predominate as compared to tose in te plane of te film. 7 Te fluid beaves as a continuum wic implies tat pressure are ig enoug so tat te mean free pat of te molecule of te fluid are muc smaller tan te effective pore diameter or any oter dimension. No slip boundary condition is applicable at te bearing surfaces. 8 Lubricant film is assumed to be isoviscous. 9 Temperature canges of te lubricant are neglected. 47

21 10 Te bearing surfaces are assumed to be perfectly rigid so tat elastic deformation of te bearing surfaces may be neglected. 11 In case of bearing working wit magnetic fluids, te lubricant is assumed to be free of carged particles. 1 Wen bearings work under te influence of electromagnetic fields, it is assumed tat te forces due to induction are small enoug to be neglected..5 MODIFIED REYNOLDS EQUATION Te differential equation wic is developed by making use of te assumptions of ydrodynamic lubrication in equations of motion and continuity equation and combining tem into a single equation governing lubricant pressure is called Reynolds equation. Te Reynolds equation wen derived for more general situations like porous bearings or ydro magnetic bearings or bearings working wit non-newtonian or magnetic lubricant, etc. is called generalied Reynolds equation or modified Reynolds equation. Tis equation is te basic governing differential equation for te problems of ydrodynamic lubrication. Te differential equation originally derived by Reynolds (1886) is restricted to incompressible fluids. Tis owever is an unnecessary restriction, for te equation can be formulated broadly enoug to include effects of compressibility and dynamic loading. We ave called tis te generalied Reynolds equation. 48

22 Figure: A Consider tatt te upper surface of te bearing surfaces is S1 and te lower surface is S wic are in relative motion wit uniform velocities U1, V 1, W 1 W and UU, V, W respectively. Te surface S 1 and S enclose te lubricant film. Te lubricant velocities in te film region F, S1 ands are u, v, w, u1, v1, w 1 and u, v, w respectively. Lubricant pressure, in F,S 1 ands are p, P1 and P respectively. Film tickness is assumed to be a function of x. 49

23 1 Te eigt of te fluid film is very small compared to te span and lengt x, y. Tis permits us to ignore te curvature of te fluid film, suc as in te case of journal bearings, and replace rotational by translational velocities. Te lubricant is Newtonian wit constant density and viscosity. 3 Tere is no variation of pressure across te fluid film dp resulting in 0. d 4 Te flow is laminar; no vertex of flow and no turbulence occur anywere in te film. 5 No external forces act on te film. Tus, X Y Z 0 6 Fluid inertia is small compared to te viscous sear. Tese inertial forces consist of acceleration of te fluid centrifugal forces acting in curved films and fluid gravity. Tus, du dt dv dt dw dt 0 7 No slip is taken in to account at te bearing surfaces. 8 Compared wit te two velocity gradients du d and dv d, all oter velocity gradients are considered negligible. Since u and, to a lesser degree, v are te predominant 50

24 velocities and is a dimension muc smaller tan eiter x or y, te above assumption is valid. Te two velocity gradients du dand dv d can be considered sears, wile all oters are acceleration terms, and te simplification is also in line wit assumption 5. Tus, any derivatives of terms oter tan du d and dv d will be of a muc iger order and negligible. We can tus omit all derivatives wit te exception of d u d and d v d. 9 Te porous region is omogeneous and isotropic. 10 Te flow in te porous region is governed by Darcy s law: Q k 1 P 1 were Q 1, k, and P 1 are respectively velocity, permeability, viscosity and pressure of te fluid in te porous region. 11 Pressure and normal velocity components are continuous at te interface. 1 Bearing is press-fitted in a solid ousing. Te equation of motion under te assumptions stated above, takes te form 51

25 p u 0 (.5.1) x p v 0 (.5.) y and p 0 (.5.3) From equations (.5.1) and (.5.) one finds tat u p (.5.4) x v p (.5.5) y and from equation (.5.3) it is cleared tat p 0 p p x, y (.5.6) Te no-slip boundary conditions are u U at 0 1, v V 1, w W1 u U at,v V,w W By integrating equation (.5.4) twice wit te above boundary conditions we ave, 5

26 53 A x p 1 u and so B A x p 1 u Similarly we arrive at 1 1 B A y p 1 v (.5.7) We now make use of te continuity equation (..9) wit no source or sinks present and wit te state of te lubricant independent of time, te continuity equation reads as, 0 w y v x u (.5.8) Te equation for omogeneous incompressible fluid takes te form 0 q. Equivalently, 0 w y v x u (.5.9) Solutions of equations (.5.7) wit related boundary conditions are

27 U U U x p 1 u, 1 1 V V V y p 1 v (.5.10) Substituting values of u and v in equation (.5.9), one can see tat 1 1 U U U x p 1 x w 1 1 V V V y p 1 y By integrating across te film tickness i.e. from 0 to, we get 1 W W U U x dx dp 1 x 1 3 V V y y p 1 y 1 3 wic can be written by rearranging te terms as y p y x p x W W V V y U U x 1 (.5.11) Tis equation is known as Generalied Reynolds equation wic olds for incompressible fluid.

28 Figure: B In most applications we consider te upper surface as non porous and moving wit a uniform velocity U Uin te x-direction togeter wit a normal velocity W and te lower surface is stationary and as a porous facing of tickness H *. Due interfaces, to continuity of velocities at te k P1 W1 0 k P and W. Substituting tese equations in equation (.5.11) we obtain te modified Reynolds equation for porous bearings as follows. 55

29 56 y p y x p x 3 3 V V y U U x P P k (.5.1) In most of te applications of bearing systems we consider one of te surfaces as non-porous and moving wit a uniform velocity U in te x- direction togeter wit a normal velocity W. Particularly let us consider lower surface is stationary and as a porous facing of tickness * H. So, 0 U 1, U U, 0 V V 1, 0 W 1, P P 1, W W and 0 P Terefore equation (.5.1) reduces to y p y x p x P 1k W 1 x U 6 (.5.13) were te pressure P in te porous region satisfies te Laplace equation 0 P y P x P (.5.14) Using te Morgan-Cameron approximation (1957) tat wen * H is small, te pressure in te porous region can be replaced by te average pressure

30 wit respect to te bearing wall tickness and wic was extensively used by Prakas and Vi (1973), it is uncoupled by substituting P 0 P P H * x y (.5.15) Tus te modified equation is x 3 p 3 p 1kH* 1kH* x y y 6 U 1W (.5.16) x Hence te problem of finding te film pressure is reduced to te solution of equation (.5.16) wit appropriate boundary conditions. However, for te modified Reynolds equation for porous bearing in cylindrical polar coordinates (c.f. Bat (003)) we ave 1 r 3 dp 1kH * r 1 4 k H * r dr d r dr 3 r r l l r (.5.17) 0 u were u and l are angular velocities of upper and lower surfaces respectively and. r u l Neuringer and Rosensweig (1964) proposed a simple model to describe te steady flow of magnetic fluids in te presence of slowly canging external magnetic fields. Te model consists of te following equations. 57

31 . q p q M. H (.5.18) q 0. q 0 (.5.19) H 0 (.5.0) M H (.5.1) H M 0. (.5.) were 0 is te permeability of free space, M is te magnetiation vector, H is te external magnetic field and is te magnetic susceptibility of te magnetic particles. Using equations (.5.0) - (.5.) equation (.5.18) becomes 0H q. q p q (.5.3) Tis equation sows tat an extra pressure 0 H is introduced into te Navier Stokes equations wen magnetic fluid is used as a lubricant. (.5.16) as Tus te modified Reynolds equation in tis case is obtained like equation x 3 0 H 1kH * p 3 1kH * x y p y 6U 1W (.5.4) x 0 H 58

32 become Te modified Reynolds equation (.5.17) for magnetic fluids 1 r 3 d 0 H 1kH * r p 1 4 k H * r dr d r dr 3 r r l l r (.5.5) 0 u In view of te discussion before to Section.3 te generalied modified Reynolds equation for a magnetic fluid based transversely roug bearing system turns out to be 1 r r kH * r d 0 H p dr kh* u d 3 r r l l r r dr 3 (.5.6).6 MAGNETIC FLUID AS A LUBRICANT Tis century as witnessed te emergence and growt of many smart materials wic ave tremendous potential for application in most of te tings from laptop, computers to concrete bridges. One suc material is magnetic fluid/ferrofluid wic as eralded a new dimension of study for various tecnological applications. In fact, 59

33 Elmore (1938) discovered te ferrofluid. Tere were not many breaktrougs in tis area for te next few decades. Because of its torougly multi disciplinary nature, an understanding of te interrelatedness of various sciences tat act synergistically, were required for developing tis study furter leading to innovative applications. Te topics tat are linked wit ferrofluids are magnetism pysics of fluid flows and dispersal system, pysical and cemical colloidal cemistry ydro and termo mecanics, mecanical engineering and material sciences etc. By now, it is well known tat ferrofluid /magnetic fluids are colloidal suspension of surfactant coated magnetic particles in a liquid medium, were te sie of te particles are of several nanometres. Te magnetic moment of an individual particle is several orders of magnitude larger tan te magnetic moments of transition or rare eart metal ions. Suc ferrofluid enjoy a variety of usual properties for instance, wen a ferrofluid is placed near a magnetic fluid gradient te box sape of te liquid distorts. Termodynamically tese ferro colloids beave as solute molecules. Hence tey defuse and interact wit solvent or oter molecules just like macro molecules. In addition, in magnetic gradient fields tese fluids exibit increased viscosity and increased apparent density. Tese fluids wic are properly stabilised, undergo practically know aging or separation. Tey remain liquid in a magnetic field and after removal of field recover teir caracteristics. Particles witin te liquid experience a body force due to te field gradient and move troug te liquid imparting drag to it causing it to flow. Tus, te magnetic fluids can be made to move wit te elp of a magnetic field gradient, even in te regions were tere is no gravity. Tis property makes tese fluids useful in space sips wic often go in ero gravity regions. Te side tat does not face te sun can be made warmer and te side tat faces te sun cooler by using magnetic fluid. Oil based or oter lubricating fluid based magnetic 60

34 fluids can act as lubricants. Te advantage of magnetic fluid lubricant over te conventional once is tat te former can be retained at te desired location by an external magnetic field. In sealed systems as in food processing macines, contamination due to conventional lubricants can be prevented by making use of magnetic fluid lubricant. Te continuum description of magnetic fluid flow, termed as ferroydrodyanmics (FHD), was introduced by Neuringer and Rosenweig (1964). Here te field forces arose from magnetically polarisable matter subjected to applied magnetic field. In fact in 1964 Neuringer and Rosenweig proposed a simple teoretical model to describe te steady flow of magnetic fluids in te presence of slowly canging external magnetic fields. Zan and Rosenweig (1980) described te motion of magnetic fluids troug porous media under te influence of obliquely applied magnetic field. Agrawal (1986) discussed te performance of a porous inclined slider bearing under te presence of a magnetic fluid lubricant. Verma (1986) considered a magnetic fluid based squeee film performance. Bot te above studies establised tat te magnetiation introduced a positive effect on te performance of te bearing system. Te load carrying capacity was found to be increased. Bat and Deeri (1991) extended te analysis of Verma (1986) by analying te squeee film between porous annular disks using a magnetic fluid lubricant. Subsequently, Bat and Deeri (1993) dealt wit te modelling of a magnetic fluid based squeee film in curved porous circular disks. Bat and Patel (1994) considered te magnetic fluid lubrication of a porous slider bearing wit a convex pad surface. Te beaviour of porous slider bearings wit squeee film formed by a magnetic fluid was investigated by Bat and Deeri (1995). Das (1998) discussed te optimum load bearing capacity for slider bearings lubricated wit coupled stress fluids in magnetic 61

35 field. V. Bastovi (1999) et al. studied te influence of filling te air gapes wit magnetic fluid on classic electromagnetic effects (a) interaction of a conductor wit te current and te magnetic field, (b) interaction of magnetic polls of two electromagnets. Patel and Deeri (00) investigated te magnetic fluid based squeee film beavior between curved plates lying along te surfaces determined by secant function. Te effect of magnetic fluid lubrication on various types of bearing system as been discussed extensively in Bat (003). Deeri et al. (006) dealt wit te performance of a circular step bearing taking a magnetic fluid wit te lubricant. Deeri et al. (007) analyed te beaviour of a magnetic fluid based squeee film between porous circular plates considering porous matrix of variable tickness. Patel et al. (008) made an effort to observe te performance of a magnetic fluid based infinitely long ydrodynamic slider bearing. Te friction was found to be decreased at te runner plate as compared to te conventional lubricant. Patel and Deeri (011) studied te effect of slip velocity on Sliomis model based ferrofluid lubrication of a plane inclined slider bearing. Tis investigation made it clear tat magnetiation could be minimied te adverse effect of rougness up to some extent wen small values of te slip parameter were involved. Recently, Sing and Gupta (01) presented a teoretical investigation concerning te effect of ferrofluid lubrication on te dynamic caracteristic of curved slider bearing based on Sliomis model for magnetic fluid flow. It was observed tat te effect of rotation of magnetic particles improved te stiffness and damping capacities of te bearing. Magnetic fluids are widely used in loud speakers to increase teir power and improve voltage current caracteristics. Tese properties ave developed te tecnology of super finis treatment of semiconductors, ceramics and optics. Body 6

36 compatible magnetic fluids offers new means of diagnosing and treating a wide spectrum of infirmities like aneurysm blockage. Te ability to position a magnetic fluid or to levitate solid objects in a magnetic fluid by te application of magnetic field permits te development of sock and vibration control equipments like dampers, sock absorbers, vibration isolators etc. Te successful syntesis of ferrofluid as promising applications suc as use of magnetic links. Magnetic fields can artificially impart ig specific gravity in ferrofluid. Tis property is exploited for separating mixtures of industrial scrap metals suc as titanium, aluminium and inc and for sorting diamonds. Ferrofluid based viscous dampers improving te performance of stepper motors, is being used extensively in automation process suc as lens grinding, robotics, disk read, write ead actuators. Magnetic fluid and nanoparticle application to nanotecnology as been a serious researc field so far as te performance of bearing system is concerned [Markus Zan (001)]. Te use of polymer coated magnetic nanoparticle as a lubricant as attracted considerable attentions, because of its application in biomedicinal treatments and biotribology [Alessandra et al. (01), Qun et al. (01), Riggio et al. (01)]. Nanomangnetic fluid nowadays as been widely used in lubrication, sealing, printing and medical treatment [Bing Cen and Yuguang Fan (011)]. Magnetic polymer nanoparticles ave been tailored made to improve te performance of te bearing system by using plasma polymerisation metod [Neamtu et al. (004)]. Plasma coated magnetic nanoparticles are found to be used in te bearing system targeted for parmaceutical industries particularly in disease detection, controlled drug delivery, biosensors and tissue engineering [Comoucka et al. (010)]. 63

37 Recently te paper by Lin et al. (01) concerns wit te effect of convective fluid inertia forces in magnetic fluid based conical squeee film plates in te presence of external magnetic field considering Sliomis model. Here te squeee film performance improved wit larger value of te inertial parameter of fluid inertia forces, volume concentration of ferrite particles and te strengt of te applied magnetic field. In te area of medicine, magnetic fluid based actuators are developed for implantable artificial earts wic are driven by external magnetic fields. In fact, by attacing drug to te surface of ferrofluid particles nowadays one can guide te drugs to te target site in te body by te use of a magnetic field. By taking an advantage of te colour canging property of ferrofluid paved te ways for a new application of tis fluid to detect defects in ferromagnetic materials..7 SURFACE ROUGHNESS: After aving some run-in and wear te bearing surfaces tend to develop rougness. Even, sometimes te contamination of te lubricants and cemical degradation of te surfaces contribute to rougness. Te rougness appears to be random in caracter, seldom following any particular structural pattern. Terefore, te bearing surfaces are usually far from being smoot. However, in most of te teoretical studies of film lubrication it was more or less assumed tat te bearing surfaces could be represented by smoot matematical planes. But tis appears to be an unrealistic assumption particularly in bearing working wit small film tickness [Halton (1958)]. 64

38 In te earlier days various matematical metods suc as postulating a sinusoidal [Burton (1963)] variation in film ticknesses were introduced to find a more realistic representation of rubbing surfaces. Tis metod is peraps more suitable in an analysis of te influence of waviness rater tan rougness. Earlier attempts for matematical modelling of te roug bearing surfaces used te postulation by saw toot curve [Davis (1963)]. Teng and Saibel (1967.a) ave introduced stocastic concepts and ave succeeded in carrying troug an analysis of a two dimensional inclined slider bearing wit one dimensional rougness in te direction transverse to te sliding direction. However, bearing surfaces, particularly, after tey ave received some run-in and wear, seldom exibit a type of rougness approximated by tis model. Cristenson and Tonder (1969.a, 1969.b, 1970) extended and refined it furter considering te tickness (x) of te lubricant film as (x) (x) s were (x) is te mean film tickness wile s is te deviation from te mean film tickness caracteriing te random rougness of te bearing surfaces. Te deviation s is considered to be stocastic in nature and described by te probability density function f (s ), c c were c is te maximum deviation from te mean film tickness. Te mean α, te standard deviation and te parameter wic is te measure of symmetry associated wit random variable s are governed by te relations E( E s ) s s, 65

39 and E s 3 E denotes te expected value given by c E(R) Rf ( s )ds c were f s 35 1 s 3 c 0,oterwise 3, c s c. Cristenson and Tonder (1971) investigated te ydrodynamic lubrication of roug bearing surfaces aving finite widt. It was concluded tat te transverse surface rougness significantly affected te performance of te bearing system. Furter, Cristenson and Tonder (197) analyed te effect of surface rougness on te ydrodynamic lubrication of journal bearing. In 1975 Cristenson et al. obtained a generalied Reynolds equation applicable to roug surface by assuming tat te film tickness function is governing a stocastic process. Kodnir and Zilnikov (1976) obtained te solutions of steady state elastoydrodynamic problems wit rougness. Tonder (1977) gave matematical treatment of te problem of lubrication of bearing surfaces depicting two dimensional distributed uniform or isotropic rougness. Patir and Cang (1979) studied te application of flow model to lubrication between roug sliding surfaces. Prakas and Tiwari (198) discussed te effect of various types of rougness patterns on te bearing surfaces on te performance of bearing systems. Prakas and Peeken (1985) investigated te combined effect of surface rougness and elastic deformation in te ydrodynamic slider bearing system. It was establised tat tere was a strong 66

40 interaction between te rougness and elasticity. Te elasticity acted so as to decrease te rougness effects. Prajapati (199) teoretically analye te performance of squeee film beaviour between rotating porous circular plates wit a concentric circular pocket. Te porous ousing was taken to be elastically deformable wit its contact surface roug. It was found tat te introduction of te pocket caused reduced load carrying capacity. Gua (1993) investigated te effect of isotropic rougness on te performance of journal bearing. Andaria et al. (1997) analyed te longitudinal surface rougness effects on ydrodynamic lubrication of slider bearings. Interestingly, it was found tat te longitudinal rougness resulted in an overall improved performance especially wen lower values of skewness were involved. A teoretical study of squeee film lubrication between porous rectangular plates was analyed by Buruke and Nanduvinamani (1998) by considering te combined effect of anisotropic nature of te permeability and surface rougness of te plates. It was sown tat e rougness parameter affected te loci of maximum load considerably. Andaria et al. (1999) investigated te effect of transverse surface rougness on te beavior of a squeee film in a sperical bearing. It was observed tat a relatively better performance was in place wen lower values of standard deviation were involved. An effect of surface rougness on ydrodynamic lubrication of slider bearing was studied by Andaria et al. (001.a). Andaria et al. (001.b) considered te effect of longitudinal surface rougness on te beavior of squeee film in a sperical bearing. Here, te effect of variance was playing a leading role for improving te performance of te bearing system. It was establised tat te effect of skewness was more significant. Bujurke et al. (007) investigated te squeee film lubrication between curved annular plates taking into account te adverse effect of transverse surface rougness. 67

41 Bujurke and Kudenatti (007) discussed te surface rougness effect on te squeee film beaviour between two rectangular plates wit an electrically conduction fluid in te presence of transverse magnetic field. It was observed tat te rougness and magnetic field provided a significant load carrying capacity and ensured a delayed squeeing time compared to classical case. S. Tamimnaii et al. (008) conducted a surface rougness experimental investigation and ardness by burnising on titanium alloy. Here, a low surface rougness and ig ardness was obtained for te spindle rotation, feed rate and dept of penetration. In fact, burnising is a ceepless macining process in wic a rotating roller or ball is pressed against metal piece. Te canges in surface caracteristics due to burnising caused improvement in surface rougness and ardness, wear resistance and fatigue and corrosion resistance. Stoudt et al. (009) discussed te fundamental relationsip between deformation induced surface rougness and strain localiation in AA5754. Here, 3 dimensional, matrix based statistical analysis was developed and integrated wit ig resolution topograpical imagine to assess te micro structural influence te evolution of plastic deformation and strain localiation in a commercial AA5754-O aluminium seet in tree in-plane strain modes. Tis study clearly demonstrates tat an accurate and straigt forward probabilistic expression tat captures te microstructural subtleties produced by plastic deformation can be developed from rigorous analyses of row topograpic data. Vladislav et al. (011) conducted a roug surface contact analysis by means of te finite element metod and of a new reduced model. M. Sedlacek et al. (01) confirmed te seminal role of skewness and kurtosis parameter describing te tribological properties of contact surfaces especially pointing out teir applications in surface texturing. Te experimental analysis was focused on reducing 68

42 friction in lubricated sliding macines. Nanduvinamani et al. (01) teoretically analye te problem of magneto-ydrodynamic couple stressed squeee film lubrication between roug circular stepped plates. It was found tat aimutal rougness pattern increased te mean load carrying capacity and squeee film time. Recently Lin (01) discussed te surface rougness effects of transverse patterns on Hopf bifurcation beaviours of sort journal bearings. It was sown tat te effect of transverse surface rougness provided a reduced subcritical Hopf bifurcation region as well as an increased supercritical Hopf bifurcation region. Siddangouda (01) studied te effect of surface rougness on te performance caracteristics of a porous inclined stepped composite bearing lubricated wit micro polor fluid. It was sown tat te load carrying capacity increased due to negatively skewed surface rougness pattern. Recently tere as been an increasing interest in te lubricants wit variable viscosity as bearing operations in macine are subjected to ig speeds, load, increasing mecanical seering process and continuous increasing pressures. Two important materials; Bingam materials and casson fluid wic are caracterised by a yield value ave been assuming considerable importance for teir applications in polymer industry, termal reactors and in bio-mecanics. (A casson fluid is a sear ting liquid wic as an infinite viscosity at ero rate of sear, yield stress below wic no flow occurs and a ero viscosity at infinite rate of sear) Batra and Kandasamy (1989) investigated te problem of rotating circular trust bearing wit casson fluid as a lubricant. It was found tat te load carrying capacity decreased and te friction increased due to casson fluid. Sa and Bat (000) considered te squeee film beaviour between rotating circular plates wen te curved upper plate wit a uniform porous facing approaced 69

43 te impermeable and flat lower plate taking a magnetic fluid lubricant in te presence of an external magnetic field oblique to te lower plate. Te increase in te load carrying capacity was mainly due to magnetiation. However, te increase in response time was depended on fluid inertia and speed of rotation of te plates besides magnetiation. Amad and Sing (007) considered a teoretical model for a magnetic fluid based porous inclined slider bearing for analying te velocity slip effect on te load carrying capacity of te bearing system. It was sown tat te slip coefficient deserved to be kept at minimum to arrive at an overall improve performance of te bearing system. Karadere (010) obtained a pressure distribution in a trust bearing by using Reynold s equation for te case of stable lubricant viscosity and isotermal conditions. Te deformation was estimated by applying te constitutive equations for te linear elastic materials to bot pad and runner. It was sown tat maximum load carrying capacity loss occurred in te steel runner bron pad pair as 3.03%. Also for a fixed load wen bearing dimension was small but deformations were large, te load capacity loss due to runner deformation is nearly of te same order as tose caused by pad deformations. Oladeinde and Akpobi (010) presented a matematical modelling regarding te ydrodynamic lubrication of finite slider bearings wit velocity slip and coupled stress lubricant. It was establised tat in order to augment te bearing performance 70