NOTES DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Te lecture presents te derivation of te Renolds equation of classical lubrication teor. Consider a liquid flowing troug a tin film region separated b two closel spaced moving surfaces. Te fluid pressure does not var across te film tickness and fluid inertia effects are ignored. From te momentum transport and continuit equations, te analsis leads to a single elliptic differential equation, namel Renolds Eqn., for te generation of drodnamic pressure in te film flow region. ppropriate boundar conditions, eiter pressure or flow conditions known, are noted for solution of Renolds Eqn. brief description of lubricant cavitation follows. Formulas for fluid mean velocities and wall sear stress differences are also derived. ppendices detail te one-dimensional fluid flow analsis of pressure generation and load capacit in plane slider bearings, Raleig step bearings and simple squeee film dampers. Nomenclature P Psat,,,, Film tickness Hdrodnamic pressure Liquid saturation pressure or dissolved gases saturation pressure d, d. ass flow rates per unit lengt Fluid velocities along,, directions,. ean flow velocities C L *, * Caracteristic fluid speeds along & across film tickness * * * t Time,, Coordinate sstem on plane of bearing Fluid densit verage fluid densit across film tickness Fluid absolute viscosit,,. Fluid sear stresses across film.,,. Wall sear stress differences NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés ()
Figure. depicts a tpical tin film geometr wit te {,, } as a coordinate sstem in te plane of te tin film bearing and wit te ais directed across te film tickness (,, t). Te flow of a Newtonian, inertialess, isoviscous fluid in te tin film region is governed b te following equations: continuit (mass conservation): (,) momentum transport: t (.) P (.) P (.3) wit te pressure P=ƒ(,, t) not varing across te film tickness. (,) surface velocities (,,t) L L << L,L (,,t) L Figure. Geometr of fluid film bearing ( << L, L) and flow velocities NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés ()
In te flow region of interest, te bottom surface = is stationar, wile te top surface, =(,,t), moves wit velocit components and in te and directions, respectivel. Te lubricant aderes (non-slips) to te bounding surfaces. Tus, te boundar conditions for te fluid velocities are: at,,, (.4) at,,, (.5) From simple kinematics, te normal velocit of te top surface equals to te temporal cange in film tickness () plus te spatial cange (advection) of te film tickness due to te lateral motion of te surface wit velocit, i.e. d d (.6) dt t dt t Integration of te - and -momentum transport equations across te film (-direction) is straigtforward since te pressure (P) is constant across te film tickness. Tis procedure and application of te boundar conditions lead to: P (.7) P (.8) Note tat te fluid velocities sow te superposition of two distinct effects. Te fluid moves due to an imposed pressure gradient (Poiseuille flow) and flows b a sear driven effect induced b te motion of te top surface (Couette flow). Te Poiseuille flow brings a parabolic velocit distribution across te film, wile te Couette flow results in a linear velocit profile. Figure. sows te (/) velocit profile for tree pressure gradient conditions, a=[( /)(dp/d)/] = -5, and 5, respectivel. Te first case corresponds to a pressure gradient decreasing in te direction of te sear induced flow, i.e. dp/d<, wile te second case denotes pure sear flow, i.e. dp/d=. Note tat a positive pressure gradient, dp/d >, causes a region of back flow closest to te stationar surface =. NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 3
velocit profiles velocit v/ Figure.. Dimensionless velocit profiles across film tickness. Pressure gradient varies, a=[( /)(dp/d)/] Bottom =.5 cross-film coordina te ( /) a=-5 a= a=5 Top = Dimensionless v elocit profiles across film tickness ass flow rates across te film tickness and mean flow velocities in te - and - directions are defined as: were ) d, ( ( ) d (.9) ; (.) d is an average fluid densit across te film tickness. Note tat if te fluid densit is onl a function of pressure, i.e. = (P), and since te pressure does not var across te film tickness, =. Barotropic liquids and most gas fluid flows undergoing isentropic or adiabatic or isotermal processes sow tis tpe of relationsip. On te oter and, tere are tin film flows were te fluid temperature canges dramaticall across te film tickness. In tis case not onl viscosit variations but also densit canges need be accounted for. Te analsis of eavil loaded clindrical and tilting pad bearings usuall calls for te inclusion of termal effects across te film: an energ transport equation must be used for adequate predictive analsis. Furter details on te psical aspects and implementation of termal effects are given later. Barotropic fluid: one wose material properties depend on pressure onl ND not temperature NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 4
Substitution of te velocit profiles, equations (.7-8), into equations (.9-) gives, for an isoviscous & barotropic fluid, te following mass flow rates (per unit lengt) and average velocities as : 3 3 P P ; Couette flow Poiseuille flow Poiseuille flow (.) P ; P (.) Te mass flow rates (and mean velocities) are te superposition of te pressure flow (Poiseuille flow) and te sear flow (Couette flow) components. Note tat te average sear driven fluid velocit ( )equals 5% of te (top) surface speed. Later on, in te stud of turbulence in fluid film bearings, te mean flow velocities will be renamed as bulk-flow velocit components. Integration of te mass flow conservation equation (.) across te film tickness () gives, d d d d (.3) t and using Leibni's integration formulae 3 : renders ( ) t g d d ( ) ( ) t g d g(, ) d ( ) d ( ) ( ) ( ) (.4) (.5) Te fluid viscosit is assumed uniform across te film tickness. Tis assumption is not valid in tin film flows wit significant temperature gradients across te film. 3 pplication of eqn. (.4) requires of continuit of te function () NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 5
Te non-slip boundar conditions for te velocities at te bottom and top surfaces, eqns. (.4,.5), are (), ( ) ; () ( ) ;, t, nd applied into equation (.5). Wit te definition of mass flow rates (, ), te conservation of mass equation becomes, ( ) t t t ( ) ( ) ( ) Simplifing like-terms leads to te conservation of mass equation across te film tickness: ( ) t (.6) or in terms of te mean flow velocities: t (.7) Note tat to arrive to te equations above, te common lubrication assumption C/L << is not needed. Tus, equation (.6) is valid for an tpe of fluid flow bounded between two surfaces. 3 3 P P Substitution of te mass flow rates across te film, ; Into te conservation of mass equation (.6) renders te Renolds equation of classical lubrication teor, i.e. 3 3 P P t (.8) Te terms on te rigt and side of Renolds equation represent te flow due to pressure gradients. Te left and side sows te flows induced b normal (squeee) motions of te bounding surface and te sear induced flow b te (top) surface sliding wit velocit. Tus, te fluid flow in tin film geometries is reduced a single differential equation of elliptic tpe for te drodnamic pressure field P(,, t). ppropriate boundar conditions for te pressure are required on te closure or boundaries of te flow domain as discussed later. Once te pressure field is obtained, i.e., eqn. (.8) solved, te fluid film velocit components are evaluated from equations (.7,.8), i.e. NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 6
P P, and It is of importance to evaluate te wall sear stresses (, ) at te bottom and top bearing surfaces. In te - and - directions, P P ; (.9) and at te bottom = surface and top = surface, P P ; (.) P P ; (.) Te wall sear stresses sow clearl distinct functions for te pressure and sear driven flows. Figure.3 depicts in scematic form tpical sear stress distributions due to Poiseuille and Couette tpe flows. Te wall sear stress differences are P P ; (.) and since te pressure gradients are related to te mean flow velocit components, equations (.), ten P ; P ; (.3) Te wall sear stress differences are, in terms of te mean flow velocit components, P ; P (.4) Te wall sear stress differences appear as simple functions of te mean flow velocit components. Tus, tere is no need to know wit detail te velocit profiles across te tin film. Note tat equation (.4) sows a local (quasi-static) balance of forces, pressure gradient forces NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 7
equal to te wall sear induced surface forces. Later, te formulas above will aid in te analsis of turbulent flows in tin film bearings. elocit Profiles dp/d= Sear Stress Profiles Pure Sear (Couette) Flow = dp/d< Pressure (Poiseuille) Flow dp/d<, > Sear & Pressure Flow Figure.3 elocit and sear stress profiles in a fluid film bearing NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 8
Boundar Conditions for te Renolds equation Te Renolds equation governing te generation of drodnamic pressure in te fluid film bearing is of elliptic tpe. Consequentl, appropriate boundar conditions are needed over te entire closure or boundar (, ) of te flow domain (, ). First, note tat a Newtonian fluid is a material not able to sustain tension. liquid will cavitate (evaporate) before it reaces its absolute ero pressure. Tis occurs at its saturation pressure (Psat). lso, if tere are dissolved gases (most likel air) in te liquid, tese will be released at teir saturation pressure (Pambient) and te fluid could not undergo a furter reduction in pressure. Hence, te liquid pressure needs to be greater tan its saturation pressure (> ero absolute) everwere in te flow domain. P(,, t) > in (, ) (.5) ost fluids under normal working conditions can sustain small levels of tension. Sometimes if te actions imposed on te fluid are ver fast, ten te liquid is able to support large tensile stresses over sort periods of time. Te likeliood of fluid tensile stresses is (usuall) not accounted for in classical drodnamic lubrication teor. Te brief discussion above points out to a more complicated process et to be well understood (and modeled) in fluid film bearing performance. Te distinctions made call for two different tpes of cavitation regimes: a) Gaseous cavitation wen dissolved gases in te lubricant come out of solution. Tus, P Psaturation gases Pambient (.6a) b) apor cavitation wen a pure liquid reaces its saturation pressure and evaporates (a pase cange) P Psaturation liquid (.6b) Te saturation pressure of most liquids usuall amounts to a minute fraction of one atmospere, i.e. it is ver close to ero absolute. Finall, note tat Renolds equation is not valid witin te fluid cavitation region since, for its derivation, te fluid is regarded as a single-pase component. Furtermore, no fluid pasecanges are accounted for witin te flow region wen deriving Renolds equation. later section (See Notes 6-a) describes in detail te penomenon of lubricant cavitation, including a (well accepted) sound psical model and a discussion on weter te cavitation one actuall includes lubricant vapor or released gases. Te penomenon of air entrainment in fluid film bearings, and in particular squeee film dampers, is of utmost importance under dnamic force operating conditions. Tis aspect of modern lubrication will also be considered in detail later (See Notes 3). NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés () 9
Oter tpe of fluid cavitation arises due to termal eating as te fluid flows troug tin film tickness, i.e., te fluid flases (boils) wen reacing its critical temperature. Tis condition is prevalent in man face seal applications andling volatile fluids, water included. Consider te boundar of te flow domain to be composed of two separate regions ( ). long te pressure is known (essential or Diriclett tpe boundar condition), i.e. on P P (.7) * nd, along te mass flow rate leaving troug te boundar is known, i.e., (.8) were = + 3 P (.9) wit (, ) as te components of te outward normal vector to te boundar. tpical boundar condition of tis tpe occurs wen a bearing as smmetr along te -plane (aial). Since =, ten, P at (.3) NOTES. DERITION OF THE CLSSICL REYNOLDS EQTION FOR THIN FIL FLOWS Dr. Luis San ndrés ()