Suneetha et al., International Journal of Advanced Engineering Technology

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1 Suneetha et al., International Journal of Advanced Engineering Technology Research Paper GENERALIZED REYNOLD S EQUATION FOR POWER LAW FLUIDS APPLICATION TO PARALLEL PLATES AND SPHERICAL BEARINGS SQUEEZING CONSIDERING THERMAL VARIATION P.Suneetha 1, V. Bharath Kumar 1, Prof. K. Ramakrishna Prasad 2 Address for Correspondence 1 Research Scholar, S.V. University, Tirupati , A.P, India 2 Professor of Mathematics, S.V. University, Tirupati , A.P, India ABSTRACT In this paper a generalized Reynolds equation for power law fluid is derived considering thermal variation and various special cases have been obtained, it is applied to study the squeeze film between parallel plates and spherical bearing considering thermal variation. A parameter q is introduced to seee the effects of thermal variation. It is shown that the effects of q is to decrease the load capacity and squeezing time and these factors increase due to the power law fluid factor n. KEYWORDS: Thermal Factor, Eccentricity, Consistency Indices INTRODUCTION: Lubrication is the Science of reducing friction by the application of suitable substance between the rubbing surfaces of bodies having relative motion. Substances applied to reduce friction are called Lubrication. In recent years the Power law model has been gained much attention due to its capacity of characterizing many types of lubrication such as polymer solutions and silicone fluids. Ng and Saibel have investigated the use of a modified model of Pseudoplastic Power law lubricant in the case of an inclined slider bearing and have shown that the load capacity is less than that of Newtonian fluid [1]. Tanner has studied the Nonlaw model and Newtonian lubrication theory for Power applied it to the short journal bearing [2,3]. The use of Power law fluids in squeeze films and externally pressurized conical bearing have been studied [4,5]. Shukla and prakash studied the lubrication of a rheostatic thrust bearing and showed that for the maximum load capacity the film thickness should be step function [6]. Dyson made a theoretical and experimental study using Power law lubricant for the elastohydrodynamic lubrication of rollers [7]. Safar studied the characteristics of various bearings using Power law lubricants by assuming an approximate profile in the fluid film [8]. It may be pointed out that due to the presence of additives in the base oil the viscosity or consistency of the resultant lubricant may not remain constant and it may vary across as well as along the film thickness [10,11]. It has been pointed out by Askwith et.al that the organic liquid in contact with bearing surface from a high viscous layer adjacent to it [10].The effects of viscosity variation has been studied using Newtonian lubricants and less attempts have been made to study the effects of consistency variation in the case of Non- Newtonian Power Law lubricants [9-13]. SQUEEZE FILM LUBRICATION: The squeeze film lubrication phenomenon applies to the two lubricated surfaces approaching each other with a normal velocity. The thin film of lubricant present between the two surfaces act as a cushion, which prevents the surfaces making instantaneous contact. The time required to squeeze out the lubricant depends upon the surface configuration, fluid properties, applied load and other things. In general the relation between the load carrying capacity and the rate of approach is the focal point of the most squeeze film E-ISSN analysis. Mathematical review or such process was done by various workers[14,15-18] ]. Although squeeze film lubrication has been generally understood for some time, the importance of the squeeze film bearing in space applications is lead to draw the attention of the many workers in sixties such a Gyrogimbal bearings. Many important effects have been studied carefully Pan et.al [19] studied characteristics of squeeze film bearings. Jackson considered the Inertia affects in the study of the squeeze film lubricants. Gould [20] investigated high pressures squeeze films for circular disks and considers the variation of viscosity with temperature and pressure. Beck et.al [21] considered the effects of supported mass motion in the study flat disk squeeze film bearings. Atten and Mc. Killop investigated the squeeze film between rotatory annuli [22]. The cases of squeeze films with the Non-Newtonian fluids have been also studied. In particular when the Non-Newtonian fluid in a Power Law it is required to study the effects of consistency variation. GENERALIZED REYNLODS EQUATION FOR POWER LAW LUBRICATS Presence of additives in lubricants and consideration of high temperatures and pressures during bearing running-in give rise to variation of viscosity in the lubricating film across as well as along the film thickness [23-25]. Keeping this in view, several attempts have been made in the past two decades to generalize the Reynolds equation considering viscosity variation. Dowson [26] generalized this equation by considering the variation of fluid characteristics (viscosity and density) across as well as along the film thickness. Ezzat and Rhode accounted viscosity variation through a viscosity profile considering thermal effects. There are experimental evidences to

2 support the need to consider viscosity variation in the fluid film under certain conditions. It is noted that the study of viscosity variation is generally carried out for Newtonian lubricants; scant attention has been given to this study for the case of Non-Newtonian lubricants. Prasad has employed power law model for the lubricant with additives and considered consistency variation across the film thickness by assuming that the consistency of the fluid film in the peripheral layer adjacent to the bearing surface is different from that of the central layer. Little attention has been given to study the effect of consistency variation which arise due to adsorption of additives and variation of temperature. Thus, consideration of additives and thermal effects leads to consistency variation across as well as along the film thickness. Keeping this in view, in this chapter, a generalized form of Reynolds equation is derived. GENERALIZED FORM OF REYNOLDS EQUATION Consider the symmetrical flow of a fluid of a power law lubricant between two identical rollers, each having a rolling viscosity U (fig.a). The thickness, h, of the fluid film in the lubricated contact is small compared to the radius r of the rollers. With the usual assumptions of lubrication theory, the basic equations governing the flow of a power law lubricant ( in one dimensional form) are given by (1) 0 (2) where u and v are velocities along the X and Y directions, p=p(x) is hydrodynamic pressure and m=m(x,y) and n are viscometric parameters called consistency and flow behaviour indicies, respectively. The lubricant behaviour is called pseudoplastic for n<1, diameter for n>1 and Newtonian for n=1.as the system is symmetrical about x axis, it is sufficient to consider the fluid region y 0 to determine velocities. Taking note on the symmetry, the following boundary conditions may be prescribed for u; 0 at y=0 u=u at y=h/2 (3) To determine the velocity u from eqn. (2.1), appropriate sign is to be attached to the velocity gradient. To facilitate the determination of its sign, the velocity and pressure profiles are shown qualitatively in Fig.2.2. It may be noted that the pressure becomes ambient at a point sufficiently away from the contact zone and reaches its maximum at a point x=-x*. In other words, 0 at x=-x* (4) Thus, in the region - x -x*, 0 and 0 and in the region x* x x c, 0 and 0, where x c is the point at which the film starts cavitating. Thus, the velocity distribution for the region y 0 can be obtained as u = U / / / - x -x (5) and / / u = U + / - x* x -x c (6) Now, integrating the equation of continuity (2) with the conditions V= 0 at y=0 (7) V=V h/2 at y=h/2 We have, / V h/2 = (8) Note that V h/2 is the resultant of the normal velocity V of the rollers and the normal velocity due to wedge action in the film segment, i.e., V h/2 =V + (9) Positive value of V denote the normal separation of rollers while the negative values signify the normal approach implying squeezing action. Substituting the expression for u from eqns. (5) and (6) in eqns. determining pressure: / = V+ - x -x* (10) / = -(V+ ) -x* x x c (11) / (f) = / (12) In order to evaluate the integral on the RHS of eqn. (12) we have to determine the function m=m(x,y).it has been mentioned earlier that the consistency of the lubricant layer adjacent to the bearing surface may be different from that of the central layer and it may vary with temperature as well the viscosity of all liquids, especially hydrocarbon lubricating oils decreases rapidly with increase in temperature. This variation in viscosity with temperature is of practical importance in the lubrication of many mechanical devices such as gears, cams etc. where the lubricants are to function over a wide range of temperature. There is no fundamental mathematical relationship to predict accurately the variation in viscosity with temperature. The available viscosity-temperature relations are purely empirical and the actual calculations require experimental data. In this study, it is assumed that thermal equilibrium exists and the consistency varies according to a given law. To apply this to real lubrication situations, the temperature at each point should be known. This requires a complete thermal calculation. However, a viscosity-temperature relationship can be replaced by a viscosity-film thickness relation. Such a replacement is possible in tribological conditions as it has been experimentally verified that the highest temperatures occur in zones where the film thickness is least. In view of the viscosity-film thickness relation suggested in reference, we assume that the consistency m of the power law lubricant varies according to the following law: m = (13) where is the consistency at the inlet film thickness h 1 where q, the thermal factor is measured q lies usually between 0 and 1. The value of q depends on the lubricant, the velocity of the bodies in contact, the velocity of the bodies in contact, the cooling conditions

3 and the flow of lubricant between bearing surfaces. Thus, q can be determined by completing the thermal calculations. However, in the entire study of this work, various values are prescribed for q while examining the effects of consistency variation without going into details of thermal calculations. It will be recalled that the rheological behaviour of the lubricant in the proximity of the solid surface results in an enhanced viscosity in the peripheral region. Particularly this phenomenon is evident when the additives in the lubricant are surface active and when the operating conditions of the lubricated contacts are in the elastohydrodynamic regime. This results in consistency variation across the film thickness. Keeping this in view, Prasad considered the consistency variation by assuming that the consistency varies as a step function across the film thickness. Taking these two aspects, viz. consistency variation / 1 / SPECIAL CASES: Case 1: When the sliding is zero i.e., V=0 along as well as across the film thickness into consideration, we define, now, consistency as, m = m 1 0 Y < = Km 1 < Y (14) K is the consistency ratio and a is the thickness of the peripheral layer. Now, using eqn. (14) in eqn. (12) we obtain, (f) = / / 1 / / (f o ) (15) (fo)=1-(1-k -1/n ) { 1-(1-2a/h) (2n+1)/n } (16) The value of (f) when substituted in eqns. (10) and (11) yields the generalized Reynolds eqn. which accounts for consistency variation: 0 / = - x -x*, y>0 (17) / 1 / 0 / =-( ) -x* x -x c, y>0 (18) / 1 / 0 / = / 1 / 0 / =-( Case 2: When the Squeezing is zero i.e., U=0 / 1 / 0 / 1 / 0 Case 3: When q=0 equation the equation (18) takes the form / / - x -x*, y>0 (19) ) -x* x -x c, y>0 (20) / = 0 - x -x*, y>0 (21) / =0 -x* x -x c, y>0 (22) 0 / = 0 - x -x, y>0 (23) 0 / =0 -x* x -x c, y>0 (24) Case 4: Taking consistency variation along the film, layer thickness a = 0 & K=1 then f 0 becomes 1 i.e., f 0 =1.In this eqns (17)&(18) takes the form(in squeezing) / / / = 0 - x -x*, y>0 (25) / =0 -x* x -x c, y>0 (26) SUMMARY: In this chapter a generalization Reynolds equation taking the consistency variation across as well as along the film is derived and various special cases have been obtained Now apply the above Reynolds equation to parallel plates and spherical bearing. PARALLEL PLATES In this section, we consider the flow between two parallel plates of length 2d, approaching each other normally with a velocity V due to a symmetrically placed load(fig b). The plates are separated by a film thickness 2h. with the usual assumptions of lubrication theory, the governing eqs. of motion for a power law fluid in the case of squeezing is obtained with reference to (Fig b) and putting U=0 in generalized Reynolds equation = V (27) (f 0 ) = 1 - ( 1-k -1/n ) { 1-(1-a/h) (2n+1)/n } (28) and h 1 is the initial film thickness measured at x = -d just before squeezing commences.pressure attains its maximum at x = 0, i.e., 0 at x = 0. Using this condition in the integration of eqn.(27) we get, (29)

4 Integrating again eqn. (29) using condition p = 0, at x = d, we obtain the expression for pressure p. Denoting it by p k,q we have P k,q = (30) The load capacity w k,q per unit width is given by W k,q = 2, (31) Which on using eqn. (30) becomes W k,q = (32) The squeezing time t k,q from an initial film thickness 2h1 to a subsequent film thickness 2h 2, say, is obtained by putting V= in eqn. (32) and integrating, we have t k,q = / / (33), / Considering consistency variation alomg the film thickness only (which can be obtained by taking k=1), eqn. (32) and (33) become And W 1,q = t 1,q = / (34) / /, / / / (35) To determine the effect of thermal factor q on load and response time, we define the following quantities:, =,, = (36), =,,,, (37) I 1,q = 1 (38) H = h/h 1, = a/h 1, H 2 = h 2 /h 1 (39) SPHERICAL BEARING: Consider squeezing of a power law lubricant between two eccentric spherical surfaces of radii r and R (r > R), approaching each other with a relative squeeze velocity V. The velocity is assumed to be constant in magnitude and in direction and symmetrically placed with respect to the boundaries of the system (Fig.c). The film thickness is given by h = c(1-εcosθ) where c = r-r and ε=e/r=r, ε being eccentricity ratio. Following a procedure similar to that adopted in the previous section, one can obtain the velocity u of the lubricant as u= / / / / 0, h/2 (40) u= / 0, 0 /2 (41) For a sphere the amount of lubricant passing through a conical element of the surface is given by Q= 2 Expressing eqns. (40) and (41) in polar coordinates and substituting in eqns. (42) we get Q=2 / / / (42) / (43) 1-(1-K -1/n ) 1 1 / 1 1 / (44) h 1 being the initial film thickness measured at /2 The displacement of lubricant at any point in the seat is given by Q= sin (45) From eqns. (43) and (45) we get = - R 2 (46) V=c. Since the pressure profile is symmetric about θ = 0, we shall consider the region 0 θ /2 where 0. Integrating eqns. (46) with boundary condition p(/2) = 0 we obtain the pressure. Denoting it by p k,q we have, 2 / d, 0 /2 (47) The load capacity W k,q for hemisphere is given by

5 W k,q = 2 /, B 2 = R 3 2 (49) The squeezing time for the surfaces to approach from the initial concentric position = 0 to a subsequent position, say is obtained as t k,q = /, / / / (48) d (50) As in this case the consistency variation along the film thickness is given by K=1, we get g 0 =1 as earlier H = h/c = 1-., =,, = J 1,q = =, /, /, /,, (51) (52) (53) Fig. 1: Effects of thermal factor q on load ratio p,q for parallel plates Fig 2: Variation of response time ratio p,q with thermal factor q for parallel plates Fig.3: Variation of response time ratio p,q with thermal factor q for different n in parallel plates Fig 4: Variation of response time ratio s,q with eccentric position ε for spherical bearing Fig 5: Variation of response time ratio s,q with q for different n in spherical bearing Eqn (36) and (37) are evaluated numerically and graphs have been plotted. From fig (1) (2)and (3) we can see that load capacity decreases with the increase of q and It is also increases with the increase of H. squeezing time decreases with the increase of q for different values H 2. i.e., the effects of thermal factor is to decrease, the load capacity and squeezing time. It is also seen from the Fig(3) that, squeezing time increase as n increase. The graphs of W s,q,t s,q are ploted.fig (1) & (2) are plotted for various n. it is found that from

6 Fig.(1) we see that W s,q, decreases with the increase eccentricity for different n. t s,q decreases with the increase of q for different n. i.e., the load capacity decrease due to the eccentricity and increases with n and squeezing time decrease due to the thermal factor with the increase of n. It is similarly to the results obtained in parallel plates. SUMMARY: In this chapter the Reynolds equation derived and is applied for n, squeezing between parallel plates and spherical bearing with power law lubricant. The effect of eccentricity is decreases of load capacity and the effect of thermal effect is decreases of squeezing and they increases with n NOMENCLATURE a peripheral layer thickness h film thickness h 1 inlet film thickness K consistency ratio m,m 1 consistency indices n flow behaviour indices u,v velocities along the coordinate axes U rolling velocity V normal velocity x,y coordinate axis -x* point of maximum pressure x c or x* point of cavitation Non-dimensional quantities corresponding to a c radial clearance d length of parallel plates 2h orh film thickness 2h 1,2h 2 initial and subsequent film thicknesses p hydrodynamic pressure q thermal factor r polar coordinate R radius of the circular plate,sphere, t time t s,q response time W K,q load capacity s, q load ratios Eccentricity Final eccentricity position REFERENCES: 1. Ng,C.W.and Saibal,E., Non-linear Viscosity Effects in Slider Bearing Lubricatiion;, J.Basic Engg. Tans. ASME., Vol.84,1962, P Tanner, R.I., Non-Newtonian Lubrication Theory and its Application to the short Journal Bearing, Australian Jounal of Applied Science, Vol.14,1963, P Tanner,R.I., A Short Bearing Solution for Pressure Distribution in a Non-Newtonian Libricant, J.App.Mech. Trans. ASME.,1964, P Hsu,Y.C.and Saibel,E., Slider Bearing Performance with a Non-Newtonian Lubricant, ASLE.,Trans.Vol.8, 1965, P J.B.Shukla ans Isa,M., Characteristics of Non-Newtonian Power Law Lubricants in StepBearing and Hydrostatic Step Seals, Wear,Vol.30,No.1,1974, P Shukla,J.B., and Prakash, J., The Rheostatic Thrust Bearing using Power Law Fluids as Lubricants, Jap.App.Phys. Vol.8,1969. P Dyson,A. and Wilson,A.R., Film Thickness in ElastoHydrodynamic Lubrication by Silicone Fluids,Proc. Inst.Mech.Engrs., Vol.180 (pt 3k) , P Safar,Z.S. and Shawki.G.S.A., Perfomance of Thrust Bearing Operating with Non- Newtonian Lubricating Fluids, International, Vol.12,1979, P Shukla,J.B., Kumar,S. and Chandra,P., Generalized Reynolds Equation with Slip at the Bearing Surfaces: Multi-Layer Lubrication Theory, Wear,Vol.60,1980, P Askwith,T.C., Cameron,A. and Crough,R.F., Chain Length of Additive in Relation to the Lubricant in the film and Bearing Lubrication, Proc.Roy.Soc.,Vol.291 A,1966, P Cameron,A. and Gohar,R., Theoretical and Experimental Studies of the Oil Film in Lubricated Point Contacts, Proc. Roy.Soc.,S.A.,Vol.291,1966, P Quale,E.B. and Wiltshire.F.R.,; The Performance of Hydrodynamic Film with viscosity variation Perpendicular to the Direction of Motion, ASME., Paper 72-Lub-E, Dyson,A., Thickness of Very Thinfilms in EHD Lubrication, Discussion in Faraday Trans.,1971, P Reynolds,O., On The Theory of Lubrication and its Applications to Mr.Beuuchamp Tower Experimental, Phil. Trans. Roy. Soc; London, Vol.177, Part I,1866, P D.F.Moore, A Review of Sqeeze Films, Wear, 8(1965) J.Stefan, Versuche uber die Scheinbare Adhasion, Sitzungber Math-Naturwiss. K1. Bayer. Akad. Wiss. Munchen, 69(1874) G.Ramanaiah and J.N.Dubey, Micropolar Fluid Lubricated Squeeze Films and Thrust Bearings, Wear,32(1975) G.Maiti, Compose(1972)98, Slider Bearing in Micropolar Fluid, Jpn.J.Apply.Phs.12(19773) C.H.T.Pan, S.B.Malanoski, P.H.Broussard, Jr.,and J.L.Burch, Theory an Experimental of Squeeze Film Bearings, Parti-Cylindrical Journal Bearings, Trans. ASME., D88(1966) P.Gould, Parallel Surface Squeeze Films: The Effects of Variation of Viscosity with Temperature and Pressure, Trans. ASME., F89(1967) J.V.Beck, W.G.Holiday and C.L.Strodtman, Experimental and Analysis of a Flat Disk Squeeze Film Bearing including Effects of Supported Mass Motion, Trans. ASME., F91(1969) C.M.Allen and A.A.Mc.Killop, An Investigation of Squeeze Film between Rotating Annuli, Trans. ASME.,F92 (1970) A. Cameron and R. Gohar, theoretical and experimental studies of the oil film in lubricated point contacts, Proc. Roy. Soc., Vol. 291A,1996,P D. Dowson, A generalized Reynolds equation for fluid film lubrication, Int. J. Mech. Sci. Vol.a, 1962, P T.C.Akswith, A. Cameron and R.f. Crounch, Chain length of additive in relation to the lubricants in the film and boundary lubrication, Proc. Roy. Soc., Vol. 291A, 1966, P Pincus,O. and Sternlich,B., Theory of Hydrodynamic Lubrication, Mc.Graw Hill Book Comp., INC.,London,1961.