EFFECTS OF COUPLE STRESSES ON PURE SQUEEZE EHL MOTION OF CIRCULAR CONTACTS

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1 -Tehnial Note- EFFECTS OF COUPLE STRESSES ON PURE SQUEEZE EHL MOTION OF CIRCULAR CONTACTS H.-M. Chu * W.-L. Li ** Department of Mehanial Engineering Yung-Ta Institute of Tehnology & Commere Ping-Tung, Taiwan 909, R.O.C. Department of Mehanial Engineering National Kaohsiung University of Applied Sienes Kaohsiung, Taiwan 807, R.O.C. S. Y. Hu *** Department of Marine Engineering National Kaohsiung Marine University Kaohsiung, Taiwan 811, R.O.C. ABSTRACT A method for investigating the pure squeeze ation in an isothermal elastohydrodynamially lubriation (EHL) problem, i.e., irular ontats lubriated with ouple stress fluid, was developed. A onstant load ondition was used in the alulations. The initial onditions suh as pressure profiles, normal squeeze veloities, and film shapes were obtained from the lassial hydrodynami lubriation theory at a speified large entral film thikness. The oupled transient modified Reynolds, elastiity deformation, and load equilibrium equations are solved simultaneously. The simulation results reveal that the effet of the ouple stress is equivalent to enhaning the lubriant visosity, thus enlarging the film thikness. The effet of ouple stress in thin film lubriation varies with film size. That is, the thinner the lubriating film is, the more obvious the effet of ouple stress is. For larger harateristi length, materials parameter, and load, the entral pressure, entral film thikness, and rigid separation are larger than those of smaller harateristi length under the same load. The time needed to ahieve maximum entral pressure and the Hertzian pressure inreases with inreasing harateristi length. Keywords : EHL, Squeeze film, Couple stress fluids, Miroontinuum theory. 1. INTRODUCTION When two bodies approah eah other along the normal diretion, very high pressure will be generated in the lubriating film due to the squeeze effets. Therefore, an elasti dimple will our at the enter of the ontat region. The related problems are alled transient elastohydrodynami lubriation (EHL) problems. These problems our in many mehanial elements with ontat pairs suh as gear teeth, ams /followers, piston ring/ylinder, rolling element bearings, and the strething proess of metal sheets. Christensen [1] has studied the pure squeeze film problem numerially, whih was lubriated by inompressible lubriants with visosity varying exponentially with pressure. Lee and Cheng [] developed a numerial sheme for pure squeeze EHL problems lubriated by a ompressible lubriant with visosity varying arbitrarily with pressure. Safa and Gohar [3] investigated the pure impat problems (a free falling ball impats on a plate). They used thin film transduers to measure the pressure in the ontat region during impat. Studying the pressure at the ontat enter, they found that pressure reahes two peaks during the total impat time. The first peak orresponds to the stage of impat where the impat fore reahes its maximum. Near the end of the rebound proess, immediately before the ball leaves the lubriated surfae, a sharp pressure peak was found at the ontat enter. Yang and Wen [4,5] made a numerial analysis of the ball dropping problem. They solved the ball s equation of motion to determine the ball s position during impat. Unfortunately, only the results for the beginning stage of the impat were presented. The result for the rebound stage was not shown. The primary peak was not reahed in their analysis. In addition, it seems unrealisti for using the Hertzian pressure as the initial * Assistant Professor ** Professor *** Assoiate Professor Journal of Mehanis, Vol., No. 1, Marh

2 ondition for onstant load onditions. Dowson and Wang [6] analyzed the bouning of an elasti sphere on an oily plate. The analysis is restrited to normal motion in the first instane, in order to develop the numerial proedure and to relate the overall findings to the results presented by Safa and Gohar [3]. Larsson and Höglund [7,8] also solved the same problems using the multi-grid method. They onluded that the maximum pressure in the lubriant film ould reah levels higher than those in the orresponding dry impat situation. They also gave theoretial evidene of the existene of the seondary pressure peak reported by Safa and Gohar [3]. Lee and Hamrok [9,10] studied the influene of ombined entraining and normal motion on the EHL of line ontats. However, there were still the limitations on load for high dimensionless materials parameter. They thought that a non-newtonian fluid must be onsidered to overome the numerial onvergene for high load and low entraining veloity problems. The effets of additive on the fluid rheology of lubriant reeived great researh attention. Instead of using the lassial ontinuum theory, many miro ontinuum theories [11~13] are proposed to model the flow rheology. Stokes theory [13] is the simplest theory whih aounts for the effets of ouple stresses, body ouple, and asymmetri tensors. This ouple stress model aims to examine the effets of partile sizes. This model is important for the appliations of pumping flow, i.e. liquid rystal, polymer-enhaned oil, animal blood, and artifiial fluid. Ramanaish and Sarker [14,15] studied the effets of ouple stress on the squeeze film between two parallel retangular plates with infinite lengths and that between two parallel retangular plates with finite size. Lin [16] studied the squeeze problems related to the long eentri journal bearings and spherial bearings. Mokhiamer et al. [17] and Naduvinamani et al. [18] studied the effets of ouple stress and linear elasti deformation on journal bearings and short porous journal bearings, respetively. The model applies to low load and large film thikness onditions. The effets of ouple stress on the squeeze film between a sphere and a plate in the hydrodynami regions are analyzed [19]. However, the study of the problem in the EHL irular ontat region is not valid. A method for general appliations with ouple stress fluid is developed to investigate the pure squeeze ation in an isothermal EHL spherial onjuntion. A onstant load ondition was used in the alulation. The initial onditions, suh as pressure profile, normal squeeze veloity, and film shape are obtained from the lassial hydrodynami lubriation theory at a speified large entral film thikness. The oupled transient modified Reynolds, elastiity deformation, and load equilibrium equations are solved simultaneously using the finite differene method and the Gauss-Seidel iteration method. In the light of the miro ontinuum theory, the effets of ouple stress fluids on the elasti deformation and the performane of a squeeze film are disussed under the onditions of onstant load.. THEORETICAL ANALYSIS.1 Modified Couple-Stress Reynolds Equation Aording to the Stokes miroontinuum theory [13], the field equations of an inompressible oupled stress fluid in the absene of body fores and body ouples are V =0 (1) DV ρ = p + ( μ η ) Dt V () where V is the veloity vetor, ρ is the density, p is the pressure, μ is the lassial visosity oeffiient, and η is a new material onstant with the dimension of momentum responsible for the ouple stress fluid property. Sine the ratio η/μ has the dimensions of length squared, the dimension of l = (η/μ) 1/ haraterizes the material length of ouple stress fluids, and l is assumed to be a material onstant in the present analysis. Two spheres approah one another in term of an equivalent sphere approahing a plane. Consider the squeeze film mehanism as shown in Fig. 1, an elasti sphere of radius R is approahing an infinite plate with a veloity under a onstant load. The lubriant in the system is taken to be an inompressible ouple stress fluid. Under the usual assumption of elastohydrodynami lubriation appliable to a thin film, the redued momentum equations and the ontinuity equation governing the motion of the lubriant given in polar oordinates an be obtained. Integrating the redued momentum equations by the boundary ondition, yields the veloity omponents. Substituting veloity omponents into ontinuity equation and integrating aross the film thikness with the boundary onditions of w(r,z), we an then derive the transient modified Reynolds equation in polar oordinates as: where f (, l h) = 1r r μ r t rh 3 p h 3 l l h f( l, h) = tanh h h l or in dimensionless form as: where 3 H P F( L, H) = K ( H) μ T 3 L L H FLH (, ) = tanh H H L (3) (4) (5) (6) 78 Journal of Mehanis, Vol., No. 1, Marh 006

3 h R w z plate sphere Fig. 1 Geometry of EHL of irular ontats at pure squeeze motion with ouple stress fluids The radial oordinate, r, has its origin at the enter of the ontat. The boundary onditions for Eq. (3) are:. Initial Stage pr (, t) = 0 (7a) r p(0, t) = 0 r (7b) prt (,) 0 (7) During the squeeze motion, two stages are inluded, i.e. the initial stage and the high-pressure stage. At the initial stage, the ball has ahieved the lubriant layer and begins to squeeze the lubriant film away. Sine the pressure is low, an iso-visous inompressible lubriant model is appropriate and the elasti deformation an be disregarded. In fat, it was indiated by Lee and Hamrok [4] that an initial ondition obtained from the hydrodynami solution was very useful in alulating the solution at the highpressure stage. In this stage, the transient modified Reynolds in dimensionless form an be expressed as: 3 P H = K ( H) T The film thikness an be expressed as: (8) H = H0 + (9) and the entral normal veloity is simply given as: H T H T 0 = = V0 (10) The pressure profile an be solved analytially from Reynolds equation, and is given as: KV0 P = (11) 4H For the onstant load, the instantaneous load balane equation is 1 Pd = (1) 0 3 By making use of Eqs. (11) and (1), the entral normal veloity is given as: V 4H 3K 0 0 = (13) At the initial stage, the equation governing the transient hydrodynami lubriation (HL) problem is analytially solved when a thin layer of oil initially separates a ball and a plate at pure squeeze motion. This HL solution is used as the initial ondition..3 High-Pressure Stage When the pressure inreases with time, the elasti deformation, and the effet of pressure on the visosity annot be negleted. This stage is denoted as the high-pressure stage. It is the problem of pure squeeze motion in elastohydrodynami lubriation. The oupled Reynolds, rheology, load equilibrium, and elastiity equations have to be solved numerially. The visosity of the lubriant is assumed to be the funtion of pressure only. The relationship between visosity and pressure used by Roelands [0] an be expressed as: μ= + μ + + (14) 9 z exp{( 9.67 ln 0)[ 1 ( p) ]} where μ 0 is the visosity at ambient pressure and z is the pressure-visosity index. The film thikness in a nominal point ontat elastohydrodynami onjuntion an be written as: r hr (,) t = h0 () t + +δ (,) r t (15) R To alulate the stati deformation due to pressure distribution, influene oeffiients D ij are introdued. The deformation an thus be omputed at disrete points i as a sum of the deformation ontributions from all pressure points j: n i Dij Pj j= 1 δ = (16) The dimensionless film thikness between two elasti bodies in irular ontats an be expressed as: H = H + + D P (17) n i i 0 ij j j= 1 where the influene oeffiients, D ij, are omputed Journal of Mehanis, Vol., No. 1, Marh

4 aording to Yang and Wen [5], Larsson and Höglund [10]. For the onstant load ase, the rigid separation is an unknown variable in eah time step. It an be determined by solving the transient Reynolds equation with the load balane equation. To save omputing time, the analytial solution of the first stage disussed above an be used as the initial ondition. 3. RESULTS AND DISCUSSION The proeeding formulation and system approah are used to study the problem of the normal approah between a ball and a flat plate overed by a thin layer of oil subjet to onstant load. However, this numerial proedure an be applied not only to the onstant load ase, but also to arbitrary squeeze ation, suh as load varying with time. To disuss the effets of ouple stress and elasti deformation on squeezing motions, the point ontat EHL problems are disussed under the onditions of non-isovosous, inompressible lubriant, and onstant load. Various operation onditions are disussed. The properties of lubriant and sphere are shown in Table 1. The initial falling height of the sphere is 0μm. For the onstant load ase, the rigid separation annot be determined by the equation of motion. Hene, the load equilibrium equation has to be inluded in the oupled transient Reynolds equation derived in Setion.1 and the elasti deformation equation. The rigid separation beomes one of the unknown variables. It will be simultaneously solved with nodal pressures. In this paper, at the initial stage, the elasti deformation and the inrease in fluid visosity and density with pressure are negleted. This solution has been obtained in Eqs. (11) and (13), and it will be employed as the initial ondition. This initial ondition is ompletely different from that used by Yang and Wen [5]. The upper limit of the omputational region was in the beginning hosen as, max = The entral differene tehnique in spae domain and the expliit tehnique in time domain are utilized to disretize the modified Reynolds equation. When more than half of the region was avitated, the maximum analyzed region max redues to half of its initial region, and so on, until max =.0. The grid was made up of 401 nodes, whih are evenly distributed, in every alulating domain. The Gauss-Seidel iteration method is employed to alulate the film thikness and pressure distribution at eah time step. Under the onditions of Newtonian fluid (L 0) and onstant load, the operation and initial onditions by Yang and Wen [5] are employed to solve the pure squeeze EHL motion of irular ontats problem using the present algorithm. The numerial results of entral pressure and film thikness and those obtained by Yang and Wen [5] are ompared as shown in Fig.. Good agreements are found. The disrepanies ome from the finer grids and alulation region varying with time of the present analysis. Table 1 Computational data Properties of lubriants for Eq.(14) Lubriants type Oil 1 Oil G (Material parameter) Inlet visosity of lubriant, Pa-s 4 4 Inlet density of lubriant, kg/m Pressure visosity oeffiient, 1/GPa Pressure-visosity oeffiient (Roelands) Properties of balls Redued radius, m Density of balls, kg/m Elasti modulus of balls, GPa 00 Poisson s ratio of balls 0.3 H=h/R ( e-5) Present Hmin Yang and Wen Hmin Present H Yang and Wen H Present P Yang and Wen P Time steps Central pressure P (GPa) Fig. Comparison of results obtained by Yang and Wen and those using the present method In the ase of onstant load onditions, Figs. 3 and 4 show the relative hange in the pressure distribution and film thikness for a flexible sphere approahing a lubriated flat surfae with a ouple stress fluid where 8 W = and G = It is observed from Fig. 3 that the pressure profile is quite flat at relatively large film thikness, but it beomes steeper with dereasing film thikness. When the sphere approahes the flat surfae, the pressure profile is almost onverged to the well-known Hertzian ontat pressure. As shown, the peak pressure is always kept at the enter in this study. It was found that the enter pressure gradually inreases with dereasing entral film thikness from urve 1 to urve 4 when the entral dimensionless film thikness dereases to a ertain level (between 0.8 to 0.9). After this stage, the pressure reverses its trend from urve 4 to urve 9, i.e., the peak pressure dereases with dereasing film thikness until it reahes a stage where the minimum film thikness and the squeeze veloity are almost zero. In this stage, the pressure gradient is signifiantly influened by the 80 Journal of Mehanis, Vol., No. 1, Marh 006

5 .50 t=0036 se t=0044 se t=0064 se P (1) t=0036 se, P= () t=0044 se, P= (3) t=0064 se, P=1.67 (4) t=0184 se, P=.093 (5) t=047 se, P= (6) t=109 se, P= (7) t=3960 se, P= (8) t= se, P=1.864 (9) t= se, P= Hertzian pressure P.0 t=013 se t=3960 se t=047 se t= se t=109 se t= se Fig. 3 H Dimensionless pressure distribution for an elasti sphere approahing a lubriated plate with time as parameter, where L = 0.545, 8 W = , G = 3500 (1) t=0036 se, H= () t=0044 se, H=1.565 (3) t=0064 se, H=51 (4) t=0184 se, H= (5) t=047 se, H= (6) t=109 se, H= (7) t=3960 se, H= (8) t= se, H=0.744 (9) t= se, H= Hertzian ontat Fig. 4 Dimensionless film thikness distribution for an elasti sphere approahing a lubriated plate with time as parameter, where L = 0.545, 8 W = , G = 3500 entraining motion term where the squeeze veloity beomes smaller. Sine the squeeze veloity for urve 9 is almost zero, it is interesting to find that the peak pressure and the entral film thikness for urve 9 are smaller than those in urve 4 due to the absene of the squeeze motion effet. As seen in Fig. 4, the position of minimum film thikness departs further from the enter (r = 0). As shown in Figs. 5 and 6, the dimensionless pressure distributions and dimensionless film thikness Fig. 5 Pressure distribution versus time with two different harateristi length (L = dash line, L = solid line) H t=0036 se t=013 se t=3960 se.0 t=0044 se t=047 se t= se t=0064 se t=109 se t= se Fig. 6 Film thikness distribution versus time with two different harateristi length (L = dash line, L = solid line) are plotted as time varying for both Newtonian fluids (L =, dash line) and ouple stress fluids (L = 0.545, solid line). In general, the effet of the ouple stress is equivalent to enhaning the lubriant visosity as shown in Eqs. (3) and (4). Subsequently film thikness is enlarged, so the dimensionless pressure spikes and dimensionless film thikness for ouple stress fluids are muh larger than that for Newtonian fluids under the same load. It is interesting to find ross points in Fig. 5. Before the maximum entral pressure spike of Newtonian fluid (t = 0013se), there is one ross point for the pressure distributions of these two fluids. Journal of Mehanis, Vol., No. 1, Marh

6 Near the entral region, the pressure for ouple stress is larger than that of the Newtonian fluid, and the pressure for ouple stress is smaller than that of the Newtonian fluid near the exit region. After the maximum entral pressure spike of the Newtonian fluid, there are two ross points for the pressure distributions of these two fluids. These phenomena an be explained by Eqs. (3) and (4). The equivalent visosity is μ/f. The smaller H/L is, the more obvious the effet of ouple stress is. From Fig. 6, the dimensionless film thikness for ouple stress fluids are muh larger than that for the Newtonian fluids under the same load, beause larger equivalent visosity results in greater film thikness. Figure 7 shows the pressure and film thikness at the ontat enter versus time with two different harateristi length (L) under onstant load ondition. The entral pressure inreases rapidly with time at the initial stage. Meanwhile, the entral pressure inreases quikly to a maximum. Then the entral pressure dereases slowly to near the amplitude of the wellknown Hertzian pressure with time at the final stage. This stage an be onsidered as the quasi-stati ondition. For larger harateristi length, the entral pressure, the entral and minimum film thikness are larger than those of smaller dimensionless harateristi length. The time needed to ahieve maximum entral pressure inreases with inreasing dimensionless harateristi length. Similarly, the time needed to ahieve the Hertzian pressure also inreases with inreasing dimensionless harateristi length. As shown, the effets of ouple stresses produe an inrease in the rigid separation with inreasing value of dimensionless harateristi length. Figure 8 shows that peak pressure varies with entral film thikness when loads are held onstant at all time steps. For a materials parameter (G = 3500) and a harateristi length (L = 0.545), the histories of the entral pressure are shown in urves B and D. It is found that the enter pressure is strongly dependent upon the load, i.e., higher load produes muh larger enter pressure. For a given load (w =.31N) and a harateristi length (L = 0.545), the histories of the entral pressure are shown in urves D and F. The higher materials parameter produes muh larger enter pressure. Moreover, it inreases the diffiulty in getting the onverged solution. It has been known that the visosity in the ouple stress fluid model is muh larger than that in the Newtonian fluid model. Therefore, by omparing urves A, C, and E with B, D, and F, it is interesting to note that the slope of peak pressure with dereasing entral film thikness for the Newtonian fluid model is muh larger than that in the ouple stress fluid model during the squeeze proess. Figure 9 shows the pressure and film thikness at the ontat enter versus the dimensionless harateristi length at t = 30ms, G = 3500 with different loads. As an be seen, the effets of ouple stresses produe an inrease in the entral pressure and entral film thikness with inreasing dimensionless harateristi length. The larger the load is, the greater the entral pressure and entral film thikness at t = 30ms are. h ( μ m) h 0 (L=0.545) h (L=0.545) h (L=0.545) min p (L=0.545) h (L=5.45) p (L=5.45) t (ms) h 0 h h min(l=) p (L=) (L=) (L=) p (GPa) Fig. 7 Effets of harateristi length (L = 0.545, 5.45, and 0) on the pressure and film thikness at the ontat enter under onstant load ondition p (GPa) Fig. 8 p (GPa) F C A D E B h ( μm) w=4.61n, G=3500, L= w=4.61n, G=3500, L=0.545 w=.31n, G=3500, L= w=.31n, G=3500, L=0.545 w=.31n, G=4000, L= w=.31n, G=4000, L=0.545 Variation in entral pressure with entral film thikness at t=30ms w=4.611n, p w=4.611n, h w=6.917n, p w=6.917n, h L w=6.917n, p max w=4.611n, p max 0.1 h ( m) Fig. 9 Central pressure and film thikness versus dimensionless harateristi length at t = 30ms, G = 3500 with different loads μ 8 Journal of Mehanis, Vol., No. 1, Marh 006

7 For larger load, the maximum entral pressure (whih are obtained from t = 0 to t = 30ms) inreases relatively rapidly with inreasing harateristi length. 4. CONCLUSIONS A method for general appliations with ouple stress fluid was developed to investigate the pure squeeze ation in an isothermal elastohydrodynamially lubriated spherial onjuntion. The oupled transient modified Reynolds, elastiity deformation, and load equilibrium equations are solved simultaneously using the finite differene method and the Gauss-Seidel iteration method. In the light of the miro ontinuum theory, the effets of ouple stress fluids and elasti deformation on the performane of squeeze film are proposed and disussed under the onditions of onstant load. The onlusions from the main results an be summarized as follows: 1. The effet of the ouple stress is equivalent to enhaning the lubriant visosity, thus enlarging the film thikness. The effet of ouple stress on thin film lubriation depends on the film size. That is, the thinner the lubriating film is, the more obvious the effet of ouple stress is.. For larger harateristi length, materials parameter, and load, the entral pressure, entral film thikness, and rigid separation are larger than those of smaller harateristi length under the same load. 3. The time needed to ahieve maximum entral pressure and the Hertzian pressure inreases with inreasing harateristi length. 4. The slope of peak pressure with dereasing entral film thikness for the Newtonian fluid model is muh larger than that in the ouple stress fluid model during the squeeze proess. NOMENCLATURE b referene Hertzian radius at load w (m) D ij influene oeffiients for deformation alulation E equivalent elasti modulus (Pa) G dimensionless material parameter, αe h film thikness h 0 rigid separation h entral film thikness h min minimum film thikness H dimensionless film thikness, hr/b K onstant in Reynolds equation, 8π/W l harateristi length of the ouple stress fluids, l = (η/μ) 1/ L dimensionless harateristi length of the ouple stress fluids, lr/b p pressure (Pa) p p h entral pressure (Pa) referene Hertzian pressure at load w (Pa) P dimensionless pressure, p/p h r radial oordinate (m) R ball radius (m) t time (se) T dimensionless time, te /μ 0 v 0 normal veloity of the ball s enter (m/s) V 0 dimensionless normal veloity of the ball s enter, v 0 μ 0 R/E b w load (N) W dimensionless load, w/e R dimensionless radial oordinate, r/b z pressure-visosity index α pressure-visosity oeffiient η material onstant responsible for ouple stress parameter μ visosity of lubriant (Pa-s) μ 0 visosity at ambient pressure (Pa-s) μ dimensionless visosity, μ/μ 0 ρ density of lubriant (kg m 3 ) δ elasti deformation (m) ACKNOWLEDGMENTS The authors would like to express their appreiation to the National Siene Counil (NSC-93-1-E ) in Taiwan for finanial support. REFERENCES 1. Christensen, H., The Oil Film in a Closing Gap, Pro. R. So. London, Series A, 6, pp (196).. Lee, K. M. and Cheng, H. S., The Pressure and Deformation Profiles Between Two Normally Approahing Lubriated Cylinders, ASME, J. Lubr. Tehnol., 95(3), pp (1973). 3. Safa, M. M. A. and Gohar, R., Pressure Distribution Under a Ball Impating a Thin Lubriant Layer, ASME, J. of Tribology, 108, pp (1986). 4. Yang, P. R. and Wen, S. Z., Pure Squeeze Ation in an Isothermal Elastohydrodynami Lubriated Spherial Conjuntion, Part 1: Theory and Dynami Load Results, Wear, 14, pp (1991). 5. Yang, P. R. and Wen, S. Z., Pure Squeeze Ation in an Isothermal Elastohydrodynami Lubriated Spherial Conjuntion, Part : Constant Speed and Constant Load Results, Wear, 14, pp (1991). 6. Dowson, D. and Wang, D., An Analysis of the Normal Bouning of a Solid Elasti Ball on an Oily Plate, Wear, 179, pp (1994). 7. Larsson, R. and Höglund, E., Elastohydrodynami Lubriated at Pure Squeeze Ation, Wear, 179, pp. Journal of Mehanis, Vol., No. 1, Marh

8 39 43 (1994). 8. Larsson, R. and Höglund, E., Numerial Simulation of a Ball Impating and Rebounding a Lubriated Surfae, ASME, J. of Tribology, 117, pp (1995). 9. Lee, R. T. and Hamrok, B. J., Squeeze and Entraining Motion in Nononformal Line Contats, Part I: Hydrodynami Lubriation, ASME, J. of Tribology, 111, pp. 1 8 (1989). 10. Lee, R. T. and Hamrok, B. J., Squeeze and Entraining Motion in Nononformal Line Contats, Part II: Elastohydrodynami Lubriation, ASME, J. of Tribology, 111, pp (1989). 11. Ariman, T. and Sylvester, N. D., Miroontinuum Fluid Mehanis, A Review, Int. J. Eng. Si., 11, pp (1973). 1. Ariman, T. and Sylvester, N. D., Appliations of Miroontinuum Fluid Mehanis, Int. J. Eng. Si., 1, pp (1974). 13. Stokes, V. K., Couple Stresses in Fluids, Phys. Fluids, 9, pp (1966). 14. Ramanaish, G., and Sarkar, P., Squeeze Films and Thrust Bearings Lubriated by Fluids with Couple Stress, Wear, 48, pp (1978). 15. Ramanaish, G., Squeeze Films Between Finite Plates Lubriated by Fluids with Couple Stress, Wear, 54, pp (1979). 16. Lin, J. R., Squeeze Film Charateristis of Long Partial Journal Bearings Lubriated with Couple Stress Fluids, Tribology International, 30, pp (1997). 17. Mokhiamer, U. M., Crosby, W. A., and EI-Gamal, H.A., A Study of a Journal Bearing Lubriated by Fluids with Couple Stress Considering the Elastiity of the Liner, Wear, 4, pp (1999). 18. Naduvinamani, N. B., Hiremath, P. S., and Gurubasavaraj, G., Squeeze Film Lubriation of a Short Porous Journal Bearing with Couple Stress Fluids, Tribology International, 34, pp (001). 19. Lin, J. R., Squeeze Film Charateristis Between a Sphere and a Flat Plate: Couple Stress Fluid Model, Computers and Strutures, 75, pp (000). 0. Roelands, C. J. A., Vlugter J. C., and Watermann, H. I., The Visosity Temperature Pressure Relationship of Lubriating Oils and Its Correlation with Chemial Constitution, ASME, Journal of Basi Engineering, pp (1963). (Manusript reeived November 8, 004, aepted for publiation April 6, 005.) 84 Journal of Mehanis, Vol., No. 1, Marh 006