Inertia Effects in Rheodynamic Lubrication of an Externally Pressurized Converging Thrust Bearing using Bingham Fluids

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1 Journal of Applied Fluid Mecanics, Vol. 1, No., pp , 19. Available online at ISSN , EISSN DOI: /acadpub.jafm Inertia Effects in Reodynamic Lubrication of an Externally Pressurized Converging Trust Bearing using Bingam Fluids G. Alexander Raymand and I. Jayakaran Amalraj Department of Matematics, SSN College of Engineering, Kalavakkam, Cennai, 611, India. Corresponding Autor (Received February, 18; accepted September 4, 18) ABSTRACT Trust bearing are innately developed to witstand axial load. Wen te bearing is subjected to ig speed operations, eavy load, ig stiffness etc., suggesting a cange in te design of te bearing plays a vital role in its performance. Friction is developed between te circular plates wile te bearing operates. To reduce tis friction, te bearing is lubricated wit lubricants suc as mineral oil, greases etc., Generally, lubricants are classified into two types tat is Newtonian and non-newtonian. However, non-newtonian fluids caracterized by an yield value suc as Bingam, Casson and Herscel Bulkley, are attracting te tribologists, at present. And also, te study of fluid inertia on trust bearing is required to optimize te performance of te bearings. In tis investigation, we ave ventured to analyze te performance of te bearing by considering te combined effects of fluid inertia forces and non-newtonian caracteristic wit Bingam fluid as lubricant in an externally pressurized converging circular trust bearing. Suc studies will be useful in te design of te bearing for te optimum performance using te appropriate lubricant in various macineries operating in an extreme condition in te industries. Averaging te inertia terms over film tickness and defining a modified pressure gradient, te reodynamic lubrication equation containing inertia terms as been analyzed. Using te appropriate boundary conditions and considering externally pressurized flow in narrow clearance between two converging discs is symmetric w.r.t r and z axis, te velocity distributions, te modified pressure gradient and tereby te film pressure and te load capacity of te bearing ave been obtained numerically for different values of Bingam number, Reynolds number and angle of convergence. In addition to tat, te effects of te inertia forces, non-newtonian caracteristics and angle of convergence on te bearing performances ave been discussed. Keywords: Reodynamic lubrication; Externally pressurized bearing; Herscel-bulkley lubricants; Angle of convergence; Load capacity. NOMENCLATURE B (r) p pa Q R1 R r,θ,z vc vr vz W Bingam number varying film tickness maximum film tickness pressure of te film atmosperic pressure flow ratex radius of film inlet radius of film outlet cylindrical polar co-ordinates velocity of te core region velocity component in r directionx velocity component in z directionx load carrying capacity δ(r) tickness of te yield surface sear rate η1 consistency index η yield value τ deviatoric stress components φ angle of convergence ρ density of te fluid,p,r,z non dimensional parameters of, p, r, z * * * * δ, vc, vr, v z non dimensional parameters of δ(r),vc,vr,vz 1. INTRODUCTION In te contemporary world, macinery industry immensely concentrates on te model of te bearing as well as te nature of te lubricant. To analyze te performances of te bearings adequately, it is

2 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. necessary to take into account te combined effects of fluid inertia forces and fluid viscous forces of non-newtonian lubricants. Terefore te study of lubricant inertia is procuring increasing importance. In recent times, empirical researc precisely affirms tat te use of lubricants wit variable viscosity can increase te property of lubrication to tat of lubricants wit constant viscosity. Te performance of te bearing depends on te design, lubricant under consideration and many oter fluid parameters. Usage of externally pressurized trust bearing lubricated wit non-newtonian lubricants in mecanical industries as many potential advantages. Suc advantages are seen in te researc works of many pioneering researcers like (El-Kayar et. al., 1981) and (Jaw-Ren Lin, 1999). (Amalraj et. al., 1) Using externally pressurized trust bearings lubricated wit Bingam fluid Amalraj et. al., teoretically analyzed te effects of inertia forces. Gertzos et. al., as studied te Journal bearing performances and derived relative eccentricity, pressure distribution, flow rate, friction coefficient and te angle of maximum pressure (Gertzos et. al., 8). Alexandros et. al., analyzed a creeping flow of a Bingam fluid in a lid-driven cavity and as examined te inertial effects. Tey also studied te strengts and weaknesses of finite volume metod and te papanastasiou regulation in te lid-cavity for various Bingam and Reynolds numbers (Alexandros et. al., 14). Larisa as developed an asymptotic solution for te axisymmetric squeeze flow of a viscoplastic Bingam medium by following asymptotic tecnique suggested earlier by Balmfort and Craster (1999) (Larisa Muravleva, 17). Jurczak and Falicki investigated analytically te pressure distribution for te case of externally pressurized bearing and squeeze film bearing wit roug surfaces lubricated wit nonnewtonian fluid (Jurczak and Falicki 16). Pavan and Viswanat investigated te Bingam fluid between two parallel plane annuli wit constant squeeze motion teoretically. Also, Te effect of radius of separation on core tickness, pressure distribution, and squeeze force for various values of Bingam number is analyzed (Pavan and Viswanat, 17). (Walicka et. al., 17) Te performance of curvulinear, externally pressurized, trust bearing influenced by te wall porosity and rougness surfaces lubricated by non-newtonian fluid is analyzed by Walicka et. al.,. Abdessamed analyzed te performance of te journal bearing are determined for various values of te non-newtonian fluids and te numerical results computed for dilatant and pseudo-plastic fluids (Abdessamed et. al., 1). (Walicka and Falicki 15) Te pressurized laminar flow of an electroreological fluid (ERFs) of a casson type in a narrow clearance between to fixed surfaces of revolution is examined by Walicka and Falicki. In Tribology, te effects of fluid inertia forces becomes significant. Te effects of fluid inertia in lubrication ave been analysed by a number of investigators. Kalil et. al., as investigated te effects of convective and centrifugal inertia forces, on te performance of externally pressurized conical trust bearings, under turbulence flow condition (Kalil et. al., 199). Batra and Kandasamy analyzed quantitatively te effect of inertia forces on te pressure and load carrying capacity wit reodynamic lubrication of te squeeze film bearing wit circular plates (Batra and Kandasamy, 1989). Usa and Vimala discussed inertia effects in a circular squeeze film bearing containing central air bubbles (Usa and Vimala, ). Kandasamy and Viswanat ave studied te combined effect of fluid inertia and Bingam lubricant in a squeeze film bearing under sinusoidal squeeze motion of circular plates (Kandasamy and Viswanat, 7). Walicka et. al., investigated te inertia and couple-stress effects on te pressure distribution and load-carrying capacity in a couple stress fluid flow wit te clearance of a bearing formed by two coaxial surfaces of revolution (Walicka et. al., 17). Sapour and Najiafi analyzed te effect of inertial term of viscoplastic fluid flowing troug a cannel lined wit iger compliant polymeric gel on ydroelastic stability of pressure driven flow (Sapour and Najafi, 17). Udaya et. al., investigated teoretically te effect of rotational inertia and pseudoplastic in an externally pressurized flow between parallel plates and concentric sperical surfaces (Udaya et. al., 11). Many researcers ave investigated inertial effect in externally pressurized trust bearing wit different types of fluids as lubricants, but very few works analyze te effects of fluid inertia forces and angle of convergence. Sina Roy et. al., observed te inertial effects in an externally pressurized trust bearing wit converging and diverging film using visco-elastic lubricant (Roy et. al., 199). Using te idea of convergence and axisymmetric in an externally pressurized converging trust bearing te core and velovity is analyzed by (Jayakaran and Alexander, 16). In tis researc work, te effects of inertia forces on te pressure and load carrying capacity of a reodynamic lubrication of an externally pressurized trust bearing wit an angle of convergence as been analyzed quantitatively. It as been found tat te inertia effect is significant for reodynamic lubricant in an externally pressurized converging trust bearing. Tere is an appreciable increase in te bearing performance wen Bingam lubricant is used. Furter, it as been observed tat te load capacity of te externally pressurized converging trust bearing is relatively more compared to tat of te uniform externally pressurized trust bearing wit uniform plates.. MATHEMATICAL FORMULATION OF THE PROBLEM Considering te symmetriness of te region between te circular plates of te bearing, te investigation is presented in te upper portion of te bearing. Te geometry of te converging bearing is as sown in Fig.(1) 588

3 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. p (5) z Te Eqs. () - (5) are to be solved under te following boundary conditions r at z (6) Fig. 1. Geometry of an Externally Pressurized Convergent Trust Bearing. δ r c at z (7) p p a at r R (8) Te constitutive equation of a Bingam fluid is given by r z is continuous, at τ = η (9) η 1 τij η 1 e, τ τ η 1 ij ij ij I (1) were τij are te deviatoric stress components, η1 and η are constants namely te plastic viscosity and yield value respectively, eij represents te rate of deformation components and I = eij eij is strain invariant. However, for all practical purposes, we consider te one dimensional form of Eq. (1) wic is given in Eq. (). τ η η r rz 1 z () Tere will be a region called core region were sear stress is less tan te yield stress wic moves wit te constant velocity, vc. Let te boundaries of δr δr te core be z and z as sown in Fig. () Fig.. Sape of te Core in an Externally Pressurized Trust Bearing. Applying basic assumptions of te lubrication teory for tin films and considering inertia forces, te governing equation can be expressed as Equation of Continuity: 1 z r r () r r z Equation of Momentum: τ ρ r r p rz r z r z (4) r z. SOLUTION OF THE PROBLEM Now by te metod of averaging inertia term in te Eq. (4) we get ρ r r dp τrz r z dz r z (1) dr z Using continuity Eq. () and boundary condition (6) and (7), we get ρ 1 dp τ rz rdz rdz r r dr z (11) Now, we introduce te modified pressure gradient as ρ 1 dp f rdz rdz r r dr (1) By Eqs. (11) and (1), we ave τ f rz (1) z Integrating Eq. (1), we get te modified pressure gradient τ f z C (14)Using Eq. () and boundary conditions (6) and (7), we get te velocity distribution in te flow region as f δ r 1 δ z, η1 4 δ were z and te velocity of te core region as (15) f δ c 1 δ,were z (16) 8η1 Te equation of conservation of mass for externally pressurized bearing in an integral form is given by Q 4πr v rdz (17) 589

4 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. Were Q is te flow rate per unit widt Using velocity distributions in (17) and integrating, we obtain πrf 1 δ δ Q (18) 1η1 Considering te equilibrium of an element in te yield δ δ surface z, it is found tat η f δ r (19) Hence Eq. (18) becomes 1Qη f 1 πr 1 δ δ () Elimination of pressure gradient from (19) and (), te algebric equation for determining te tickness of te yield surface δ(r) can be obtained as 1Q 1 δ δ (1) 6η πr τ δ Te variation of film tickness of te lubricant in te converging bearing can be defined as r r tan φ R () Were, (r) represents te varying film tickness between te plates, is Maximum film tickness at te center of te bearing and φ is te angle of convergence. Te following non dimensional parameters are introduced. r p r* ; δ* δ r*; p ; * ; R Qη 1 π z πr η z* ; B Qη1 () Here, B is te Bingam Number. Using te non dimensional quantities in Eqs. (15), (16) and (1), we get te velocity of te fluid in te flow region as 1 δ * z * δ * r * vr, r*1 *tanφ δ * 1 δ * δ * * * were z* Also, te velocity of te core region is * 1 v c, r* 1 r*tanφ δ * δ * * were z* (4) (5) Substituting vr and vc from Eqs. (15) and (16) in to Eq. (1) and using f from (18), te nondimensionalized pressure gradient will be of te form dp * 1 k 1k, dr * r*1 r*tanφ 1 δ * δ * Were 7δ * 8tanφ r* 1 r*tanφ δ* k1 Re dδ * 7δ * δ * 16 r * 7δ * dr * k Re r* 1 r*tanφ δ* (6) Eliminating te pressure gradient from Eqs. (19) and (), we obtain a non linear algebric equation for determining te core tickness 1 δ * δ * Br *1 r *tanφ δ* (7) were δ = δr is non-dimensional core tickness for a flow Q, B is known and ence Eq. (7) can be solve. Te roots of te te Eq. (7) determine te sape of te plug core region. Differentiation (7) w.r.t r we get δ * r *tanφ1 dδ * dr * B δ * 1 r* 1 r*tan φ (8) Te value of δ as been determined for different values of B and for various values of r*, by iterative tecnique and reported elsewere (Jayakaran and Alexander, 16). Te pressure distribution can be obtained by substituting (8) in (6) and integrating using boundary condition (8) we get, * 1 dp * P * Pa dr* r* dr * (9) ρq Were Re is te Reynolds number. πη1r Te pres sure distribution as been calculated for different values of Bingam number, Reynolds number and Angle of convergence numerically. Again, te load carrying capacity W for te externally pressurized trust bearing can be obtained by integrating te pressure over te entire region and we get W 1 P* P * * * R* a r dr () R were, R* 1 is te ratio of inside to outside R radius of te bearing. Tis integration is performed 59

5 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. numerically for various values of te Re, B, and φ. 4. RESULTS AND DISCUSSION Te radial distribution of te film pressure as been obtained for various values of Bingam number (B), Reynolds number (Re) and te Angle of convergence (φ) are sown in Figs. (), (4) and (5). Te pressure is maximum in orifice and gradually decreases to te Peripery. We observe tat te pressure tends to increase wit te increase in Bingam number, Reynolds number and Angle of convergence. However, te quantum of increase is less for ig Reynolds number. Moreover, increase in pressure is more significant wen angle of convergence is increased. Te Numerically computed results of load carrying capacity for various Bingam number, Reynolds number and Angle of convergence for different levels of inlet radius are given in te tables (1)-(4). It is observed from te analysis tat tere is an appreciable increase in te value of te load capacity for te Bingam lubricants relative to tat of Newtonian lubricant. Tis is due to te fact tat Bingam, being a tick viscous model, sows a ig load carrying capacity. Furter, te load carrying capacity of te bearing as been found to increase gradually wit te increase of Reynolds number. However, te quantum of increase is less for ig Reynolds number as sown in Figs. (6) and (7). Te angle of convergence elps to increase te load capacity of te bearing wen we increase it, up to a particular position, as sown in Figs. (8) and (9). Beyond tis critical angle (φ < ) te load carrying capacity of te bearing starts to deteriorate wit increase in Reynolds number for any B and R* as sown in Fig. (1). Wen te inlet radius R* decreases we expect te increasing trend in te workload as it s observed and sown in te Fig. (11). Te inertia plays a small role in te bearing performance. For B = 5 & B = 1, φ = to φ = 5 and R*=.5 & R*=.1 te cange in percentage of increase in te load capacity due to inertia forces are sown in tables (4)-(6). From te results it is observed tat te percentage increase in load capacity reduces for materials wit larger yield value and increases in angle of convergence. Furter, tis percentage increase is found to increase wit an increase in Reynolds Number but percentage decreases as te angle of convergence increases for any Bingam Number. Even toug te load carrying capacity of te bearing is found to increase wit an increase of Reynolds number, te increase is not muc for ig viscous fluids i.e for fluids wit ig Bingam Number. For a te particular angle of convergence (φ = ) wic corresponds to flat externally pressurized trust bearing, our results are found to be matcing wit results of (Jayakaran et. al., 1). Fig.. Pressure profile for various Reynolds Number and Angle of Convergence for B =5. Fig. 4. Pressure profile for various Reynolds Number and Angle of Convergence for B =1. Fig. 5. Pressure profile for various Reynolds Number and Angle of Convergence for B = 15. Fig. 6. Load carrying capacity for various Reynolds Number & Bingam Number for φ=5. Fig. 7. Load carrying capacity for various Reynolds Number & Bingam Number for φ =

6 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. Fig. 8. Load carrying capacity for various Angle of Convergence. Table Load Capacity for different value of Re and Angle of Convergence for B = 5 and R =.1 B=5 Reynolds number ( 1 1 ) 1 4 φ= φ= φ= φ= φ= φ= Fig. 9. Load carrying capacity for various Reynolds Number and φ for B = 5 wit Inlet Radius R =.1. Table Load Capacity for different value of Re and Angle of Convergence for B=1 and R=.1 B=1 Reynolds number ( 1 1 ) 1 4 φ= φ= φ= φ= φ= φ= Fig. 1. Load carrying capacity for various Reynolds Number and φ > for B = 5 wit Inlet Radius R =.1. Table 4 Cange in Percentage in load Capacity wit respect to Re= for B = 5 wen R =.5 B=5 Reynolds number ( 1 1 ) φ= φ= φ= φ= φ= φ= Fig. 11. Load carrying capacity for various Reynolds Number, Bingam Number and Inlet Radius for φ = 5. Table 1 Load Capacity for different value of Re and Angle of Convergence for B = 5 and R =.5 B=5 Reynolds number ( 1 1 ) 1 4 φ= φ= φ= φ= φ= φ= Table 5 Cange in Percentage in load Capacity wit respect to Re= for B = 5 wen R =.1 B=5 Reynolds number ( 1 1 ) φ= φ= φ= φ= φ= φ= Table 6 Cange in Percentage in load Capacity wit respect to Re= for B = 1 wen R =.1 B=1 Reynolds number ( 1 1 ) φ= φ= φ= φ= φ= φ=

7 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , CONCLUSION Te above investigations reveals tat te increase in Bingam number, Reynolds number and te Angle of convergence significantly enances te bearing performance suc as pressure distribution and load carrying capacity. However te effect of fluid inertia on te bearing performance is found to be significant wen te inlet is small. ACKNOWLEDGMENTS Te autors tank te Principal and management of SSN institutions for teir support in carrying out tis work. REFERENCES Abdessamed N., S. Larbi, H. Belanece, and M. Malki (1). Journal Bearings Lubrication Aspect Analysis Using Non-Newtonian Fluids. Advances in Tribology, 1-9. Alexandros S., G. C. Georgiou and A. N. Alexandrou (14). Performance of te finite volume metod in solving regularised Bingam flows: Inertia effects in te lid-driven cavity flow. Journal of Non-Newtonian Fluid Mecanics, 8-9, Amalraj, I. J., S. Narasimman and A. Kandasamy (1). Inertia Effects in Reodynamic Lubrication of an Externally Pressurized Trust Bearing Using Bingam Lubricant wit Sinusoidal Injection. Journal of Applied Fluid Mecanics 6(4), Balmfort, N. J. and R. V. Craster (1999). Ocean waves and ice seets, J. Fluid Mec. 95, Batra, R. L. and A. Kandasamy (1989). Inertia effects in reodynamic lubrication of a squeeze film bearing. Wear, Elsevier Sequoia S. A., (in Neterlands), 7-8. El-Kayar, A., E. A. Salem and M. F. Kalil (1981). Beaviour of Externally Pressurized Conical Bearings Lubricated Wit non-newtonian Fluids. Wear, Elsevier Sequoia S.A., Lausanne (in Neterlands), Gertzos, K. P., P. G. Nikolakopoulos and C. A. Papadopoulos (8). CFD analysis of journal bearing ydrodynamic lubrication by Bingam lubricant. Tribology International 41, Jafargolinejad S. and M. Najafi. (17). Inertia flows of Bingam fluids troug a planar cannel: Hydroelastic instability analysis. Journal of Mecanical Engineering Science (1), Jaw-Ren L. (1999). Static and dynamic caracteristics of externally pressurized circular step trust bearings lubricated wit couple stress fluids. Tribology International, Jayakaran Amalraj, I. and G. Alexander Raymand (16). Core Variation in an Externally Pressurized Converging Trust Bearing wit Bingam Lubricant. Applied Mecanics and Materials 85, Jayakaran A., I., S. Narasimman and A. Kandasamy (1). Inertia Effects in Reodynamic Lubrication of an Externally Pressurized Trust Bearing using Bingam Fluids. 8 t Asian Computational Fluid Dymanics Conference, Hong Kong SAR, (in Cina). Jurczak, P. and J. Falicki (16). Pressure Distribution in a Squeeze Film Sperical Bearing wit Roug Surfaces Lubricated by an Ellis Fluid. Int. J. of Applied Mecanics and Engineering 1(), Kandasamy, A. and K. P. Viswanat (7). Reodynamic Lubrication of a squeeze film bearing under sinusoidal squeeze motion. Computational and Applied matematics, 6, Kalil, M. F., S. Z. Kassab and A. S. Ismail (199). Effect of inertia forces on te performance of externally pressurized conical trust bearing under turbulent flow condition. Wear 166(), Larisa M. (17). Axisymmetric squeeze flow of a viscoplastic Bingam medium. Journal of Non-Newtonian Fluid Mecanics 49, Singeetam P. K. and K. P. Viswanat (17). Squeezing of Bingam Fluid Between Two Plane Annuli. Applications of Fluid Dynamics, Lecture Notes in Mecanical Engineering, Sina Roy, J., S. Pady and L. K. Bopa (199). Inertia effect in externally pressurized trust bearing wit converging and diverging film using visco-elastic lubricant. Actamecanica, Springer-Verlag 96, 1-1. Udaya, P. S., R. S. Gupta and V. K. Kapur (11). Effects Of Inertia In Te Steady State Pressurised Flow Of A Non-Newtonian Fluid Between Two Curvilinear Surfaces Of Revolution: Rabinowitsc Fluid Model. Cemical and Process Engineering (4), Usa, R. and P. Vimala (). Inertia effects in circular squeeze film containing a central air bubble. Fluid Dynamics Researc 6, Walicka, A. and J. Falicki (15). Reynolds number effects in te flow of an electroreological fluid of a Casson type between fixed surfaces of revolution. Applied Matematics and Computation 5, Walicka, A., E. Walicki, P. Jurczak and J. Falicki 59

8 G. Alexander Raymand and I. Jayakaran Amalraj / JAFM, Vol. 1, No., pp , 19. (17). Influence of Wall Porosity and Surfaces Rougness on te Steady Performance of an Externally Pressurized Hydrostatic Conical Bearing Lubricated by Rabinowitsc Fluid. J. of Applied Mecanics and Engineering (), Walicka, A., P. Jurczak and J. Falicki (17). Inertia and Couple-Stress Effects in a Curvilinear Trust Hydrostatic Bearing. J. of Applied Mecanics and Engineering (),