International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 http://www.sciencepublishinggroup.com/j/ijmea doi:.648/j.ijmea.75.4 ISSN: -X (Print); ISSN: -48 (Online) Non-newtonian Rabinowitsch Fluid Eects on the Lubrication Perormances o Sine Film Thrust Bearings Jaw-Ren Lin, *, siu-lu Chiang, Yu-Kang Chiu Department o Mechanical Engineering, Nanya Institute o Technology, Taoyuan, Taiwan Master Program o Mechanical and Mechatronics Engineering, Nanya Institute o Technology, Taoyuan, Taiwan Email address: jrlin@nanya.edu.tw (Jaw-Ren Lin), rotex@nanya.edu.tw (siu-lu Chiang), abba88@gmail.com (Yu-Kang Chiu) * Corresponding author To cite this article: Jaw-Ren Lin, siu-lu Chiang, Yu-Kang Chiu. Non-newtonian Rabinowitsch Fluid Eects on the Lubrication Perormances o Sine Film Thrust Bearings. International Journal o Mechanical Engineering and Applications. Vol. 5, No., 7, pp. 6-67. doi:.648/j.ijmea.75.4 Received: March 7, 7; Accepted: May 5, 7; Published: May, 7 Abstract: Thrust bearings are generally developed to sustain axial load generated rom rotating machinery. Predicting lubrication perormances o thrust bearings plays an important role in engineering applications. Taking into account the eects o non-newtonian rheology, the lubrication perormances or a sine ilm thrust bearing are investigated in the present paper. By employing a small perturbation technique to the nonlinear non-newtonian Reynolds equation, analytical solutions o the ilm pressure, the load-carrying capacity, the riction parameter and the required volume low rate are derived. Comparing with the case lubricated with a Newtonian lubricant, the non-newtonian Rabinowitsch luids (including the pseudo-plastic luids and the dilatant luids) show signiicant inluences on the lubrication perormances o a sine ilm thrust bearing, especially or a larger value o the non-newtonian parameter. Keywords: Non-newtonian Rheology, Sine Film Thrust Bearings, Load Perormances. Introduction Thrust bearings are generally developed to sustain axial load generated rom rotating machinery. Predicting load perormances o thrust bearings plays an important role in engineering applications. The load perormances o thrust bearings lubricated with a Newtonian lubricant have been investigated by many researchers, or example, the studies by Williams [], amrock [], Taylor and Dowson [], Talmage and Carpino [4], Lin et al. [5], and Lin and ung [6]. Owing to the development o modern engineering, the increasing use o a Newtonian lubricant blended with small amounts o long chained polymer solutions has gained great attention. owever, the low behavior o these kinds o non-newtonian lubricants displays a nonlinear relationship between the shearing stress and the shearing strain rate. From the experimental works by Wada and ayashi [7], this nonlinear relationship can be simulated by an empirical cubic stress model, or the so-called Rabinowitsch luid model. In this non-newtonian luid model, the nonlinear relationship or one dimensional low between the shearing stress τ and the shearing strain rate u / y holds as the ollowing equation. u y τ + n τ µ In this expression, strain rate u / y denotes the initial viscosity o a Newtonian lubricant, and n represents a nonlinear actor characterizing the non-newtonian rheology. This non-newtonian Rabinowitsch luid model is applicable to dilatant lubricants or n <, to Newtonian lubricants or n, and to pseudo-plastic lubricants or n >. Many studies have applied this model to investigate the thin ilm lubrication systems, or example, the squeeze ilm bearings by Lin [8] and su et al. [9], the hydrostatic bearings by Sharma et al. [], the journal bearings by Bourging and Gay [], and the inclined plane slider bearings by Lin []. According to their results, the inluences o non-newtonian Rabinowitsch luids on the lubrication perormances o the squeeze ilm bearings, the hydrostatic bearings, the journal bearings and the inclined plane slider bearings are signiicantly apparent. Since understanding the lubrication ()
International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 6 perormances o thrust bearings is helpul in designing the bearing system. In order to provide more inormation or the application o non-newtonian luids on lubrication ields, the study o non-newtonian eects o Rabinowitsch luids on the lubrication perormances o sine ilm thrust bearings is thereore motivated. According to the Rabinowitsch luid model, the present paper is to investigate the non-newtonian eects on the lubrication perormances o sine ilm thrust bearings. By employing a small perturbation technique to the nonlinear non-newtonian Reynolds equation, analytical expressions o the ilm pressure, the load capacity, the riction parameter and the required volume low rate are obtained. Comparing with the case lubricated with a Newtonian lubricant, the lubrication perormances o sine ilm thrust bearings with a non-newtonian Rabinowitsch luid are presented and discussed through the variation o the non-newtonian parameter and the inlet to outlet ilm ratio.. Analysis Figure shows the physical geometry o a sine ilm thrust bearing lubricated with a non-newtonian luid. The bearing length in the x -direction is B and the bearing width perpendicular to the xz -plane is B. In the present study, the bearing is considered to be ininitely wide, i.e., B >> B. The inlet ilm height in the y -direction is h, the outlet ilm height is h, and the sliding velocity in the x -direction is V s. The local ilm height h or the sine ilm bearing can be described by h( x) h + h ( x) () sin where the sine curve hsin ( x ) can be expressed as: πx hsin ( x) ( h h ) sin B Assume that the thin-ilm lubrication theory o Williams [] and amrock [] is applicable, then the incompressible continuity equation and the momentum equations and can be written as ollows. u v + x y τ p y x p y where u and v are the velocity components in the x and y directions respectively, and p is the ilm pressure. () (4) (5) (6) Figure. Physical geometry o a sine ilm thrust bearing lubricated with a non-newtonian luid. The boundary conditions o the velocity in the x direction are: u V s at y (7) u at y h (8) Solving equations () and equation (5) with the related boundary conditions, one can obtain the expression o the velocity component. where u A u u + u (9) p y hy x µ p 4 + n y hy + h y h y x 4 8 8 y h [ + n ( p / x) h / 4] ub Vs n ( p / x) ( y hy / + h y / 4) h [ + n ( p / x) h / 4] A B () () The boundary conditions o the velocity in the y direction are: v at y () v at y h () The integrated orm o the continuity equation (4) across the local ilm height is h u h v dy + dy x y (4) y y Applying the related boundary conditions, one can derive a nonlinear non-newtonian Reynolds equation or the sine ilm bearing.
64 Jaw-Ren Lin et al.: Non-newtonian Rabinowitsch Fluid Eects on the Lubrication Perormances o Sine Film Thrust Bearings d dp dp h nh Vs dx dx dx dx 5 dhsin + µ (5) P at X (6) Solving these two dierential equations, one can obtain This Reynolds equation can be used to determine the inluences o non-newtonian rheology on the ilm pressure and thereater the load-carrying capacity o the sine ilm thrust bearing. In order to conveniently analyze the bearing perormances, the non-dimensional variables and parameters are introduced as ollows. where 6( ) 6( ) P + (7) N 5 (8) X sin ( ) X sin x h ph X,, P (6) B h µ V B s µ, N h R h n V s (7) h hsin πx ( R ) sin h (8) Then the non-dimensional Reynolds equation can be expressed as: d dp N dp + d 5 sin (9) It is diicult to obtain the ilm pressure rom this nonlinear non-newtonian Reynolds equation. Apply a small perturbation method to the ilm pressure P operating at small values o the nonlinear non-newtonian parameter, << N << +, then: P P + N P + O( N ) () Substituting this expansion into the non-dimensional Reynolds equation (9) and collecting the order O( N ) and the order O( N ), thereater one can obtain two coupled equations responsible or the zero order pressure and the irst order pressure, respectively. d dp πx π ( R ) cos d dp 5 dp + N The boundary conditions or the ilm pressure are: () () P at () P at (4) P at X (5) X X ( X ) (9) X ( X ) () sin ( X ) () X ( X ) () ( X ) () Integrating the ilm pressure, one can obtain the load-carrying capacity, w. B w pb dx (4) x Introduce the non-dimensional load carrying capacity, W. wh VsB B W (5) µ Ater perorming the integration, one can obtain the non-dimensional load carrying capacity. where 6( g g ) 6( g ) W + (6) N 5 g (7) g g (8) (9) The values o the non-dimensional load carrying capacity W can be obtained by the method o numerical integration. The riction orce acting at the lower surace is
International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 65 B τ x y Introduce the non-dimensional variable, B dx (4) reduce the bearing load; however, the eects o non-newtonian dilatant lubricants ( N.5 ; N.5 ) provide an increase in the load capacity o the sine ilm thrust bearing. F h (4) µ V BB s Then the non-dimensional riction orce can be expressed as: F N [ ( dp / ) / 4] ) (4) dp + The riction coeicient is c (4) w Applying the non-dimensional riction orce and the non-dimensional load capacity, one can obtain the non-dimensional riction parameter F. B F F c (44) h W Figure. Load capacity W as a unction the inlet-outlet ilm ratio R or dierent N. Integrating the velocity ield, one can obtain required volume low rate q. h u Bdy (45) y q Introduce the non-dimensional required low rate Q. q Q (46) V hb s Ater perorming the integration, one can obtain the non-dimensional required volume low rate. 5 dp dp Q + N 8 (47). Results and Discussion In order to present the results, the non-dimensional load capacity are illustrated or the bearing operating at the inlet to outlet ilm ratio, R ~ 4.5 and the non-newtonian parameter, N.5 ~ +.5. Figure presents the non-dimensional load capacity W as a unction o the inlet-outlet ilm ratio R or dierent values o the non-newtonian parameter N. Comparing with the case o Newtonian lubricant ( N ), the inluences o non-newtonian pseudo-plastic luids ( N.5 ; N.5 ) Figure. Load capacity W as a unction o the non-newtonian parameter N or dierent R. Figure shows the non-dimensional load capacity W as a unction o the non-newtonian parameter N or dierent values o the inlet-outlet ilm ratio R. It is observed that the values o the load capacity W decrease with increasing values o the non-newtonian parameter N rom negative values ( N <, non-newtonian dilatant lubricants) to zero ( N, Newtonian lubricants), and then to positive values ( N >, non-newtonian pseudo-plastic lubricants).
66 Jaw-Ren Lin et al.: Non-newtonian Rabinowitsch Fluid Eects on the Lubrication Perormances o Sine Film Thrust Bearings Figure 4. Friction parameter F as a unction the inlet-outlet ilm ratio R or dierent N. Figure 4 describes the riction parameter F as a unction o the inlet-outlet ilm ratio R or dierent values o the non-newtonian parameter N. It is observed that the riction parameter decreases with increasing values o the inlet-outlet ilm ratio. Comparing with the Newtonian-lubricant case (N ), the eects o non-newtonian pseudo-plastic luids ( N.5 ; N.5 ) decrease the riction parameter. But, the inluences o non-newtonian dilatant behavior ( N.5 ; N.5 ) yield higher values o the riction parameter or sine ilm thrust bearings. Figure 5 shows the riction parameter F as a unction o the non-newtonian parameter N or dierent values o the inlet-outlet ilm ratio R. It is also observed that the values o the riction parameter decrease with increasing values o the non-newtonian parameter rom the region o non-newtonian dilatant lubricants to a Newtonian lubricant, and then to the region o non-newtonian pseudo-plastic lubricants. Figure 6. Volume low rate Q as a unction the inlet-outlet ilm ratio R or dierent N. Figure 6 depicts the required volume low rate Q as a unction o the inlet-outlet ilm ratio R or dierent values o the non-newtonian parameter N. It is observed that the required volume low rate or sine ilm thrust bearings increases with increasing values o the inlet-outlet ilm ratio. It is observed that the inluences o non-newtonian lubricants on the required volume low rate are slight. Figure 7. Volume low rate Q as a unction the non-newtonian parameter N or dierent R. Figure 5. Friction parameter F as a unction the non-newtonian parameter N or dierent R. Figure 7 shows the required volume low rate Q as a unction o the non-newtonian parameter N or dierent values o the inlet-outlet ilm ratio R. It is observed that the values o the required volume low rate increase slightly with increasing values o the non-newtonian parameter rom the region o non-newtonian dilatant lubricants to a Newtonian
International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 67 lubricant, and then to the region o non-newtonian pseudo-plastic lubricants. 4. Conclusion Taking into account the eects o non-newtonian rheology, the lubrication perormances or a sine ilm thrust bearing are investigated in the present paper. Based upon the Rabinowitsch luid model, a nonlinear non-newtonian Reynolds equation is obtained or a sine ilm thrust bearing. By employing a small perturbation technique to the nonlinear Reynolds equation, analytical expressions o the ilm pressure, the load capacity, the riction parameter, and the required volume low rate are derived. Comparing with the sine ilm bearing with a Newtonian lubricant, the inluences o non-newtonian pseudo-plastic luids decrease the load capacity and the riction parameter, and increase the required volume low rate. owever, the eects o non-newtonian dilatant lubricants provide an increase in the load capacity and the riction parameter, as well as a reduction in the required volume low rate or the sine ilm thrust bearings. Reerences [] C. M. Taylor and D. Dowson, Turbulent lubrication theory - application to design, ASME Journal o Lubrication Technology, 974, Vol. 96, pp. 6-46. [] B. J. amrock, Fundamentals o Fluid Film Lubrication. McGraw-ill: New York, 994. [] J. A. Williams, Engineering Tribology. Oxord University Press Inc.: New York, 994. [4] G. Talmage and M. Carpino, A pseudospectral-inite dierence analysis o an ininitely wide slider bearing with thermal and inertia eects, STLE Tribology Transactions, 997, Vol. 4, pp. 5-58. [5] J. R. Lin, R. F. Lu, and C. B. Yang, Linear stability analysis o a wide inclined plane slider bearing, Journal o Science and Technology,, Vol., pp. 49-54. [6] J. R. Lin and C. R. ung, Analysis o dynamic characteristics or wide slider bearings with an exponential ilm proile, Journal o Marine Science and Technology, 4, Vol., pp. 7-. [7] S. Wada and. ayashi, ydrodynamic lubrication o journal bearings by pseudoplastic lubricants (Part, experimental studies), Bulletin o the JSME, 97, Vol. 4, pp. 79-86. [8] J. R. Lin, Non-Newtonian squeeze ilm characteristics between parallel annular disks: Rabinowitsch luid model, Tribology International,, Vol.5, pp. 9-94. [9] C.. su, J. R. Lin, L. J. Mou, and C. C. Kuo, Squeeze ilm characteristics o conical bearings operating with non-newtonian lubricants Rabinowitsch luid model, Industrial Lubrication and Tribolog, 6, Vol. 66, pp. 7-78. [] S. C. Sharma, S. C. Jain and P. L. Sah, Eect o non-newtonian behaviour o lubricant and bearing lexibility on the perormance o slot-entry journal bearing, Tribology International,, Vol., pp. 57-57. [] P. Bourging and B. Gay, Determination o the load capacity o inite width journal bearings by inite element method in the case o a non-newtonian lubricant, ASME Journal o Tribology, 984, Vol. 6, pp. 85-9. [] J. R. Lin, Non-Newtonian eects on the dynamic characteristics o one-dimensional slider bearings: Rabinowitsch model, Tribology Letters,, Vol., pp. 7-4.