The present theoretical study is concerned with the analysis of surface roughness effects on the steady state performance of stepped circular hydrostatic thrust bearings lubricated with non -Newtonian Rabinowitsch fluids . To take the effects of su rface roughness into account, Christensen theory of rough surface has been adopted. Expression for pressure gradient has been derived by means of energy integral approach. Results for film pressure, load carrying capacity and lubricant flow rate has been p lotted and analysed on the basis of numerical results. Significant variations in the theoretical results of these properties have been observed. KEYWORDS Hydrostatic lubrication; pressurized bearings; thrust bearings; surface roughness, cubic stress fluids INTRODUCTION Amongst the e xternally pressurized bearing s, circular hydrostatic thrust bearings are of great importan ce for industrial and engineering applications due to its wide uses in high speed rotating machines. Researchers have also paid significa nt attention towards improvement of performance of these bearings under various operating and lubricating conditions [1–5]. Several investigations have been carried out to analyse the dependence of different performance properties of these be arings on rotational inertia, fluid compressibility and temperature variation of the fluid [6–8]. Optimization of design of hydrostatic thrust bearings with special consideratio n on shape and radii of recess and supply hole , and different operating conditions have been done time -to-time [9–14] . Yadav and Kapur [15] considered the combined effect of temperature and inertia on the overall performance of hydrostatic step thrust bearings. Some investigators also focussed their attention to analyse the d ynamic performance of hydrostatic thrust bearings lubricated with Newtonian and couple stress fluid s [16,17] . On the same time, many researchers also emphasized that the surface roughness plays very important role on the performance of lubricated bearings [18]. Prakash and Tiwari [19] presented the analysis of porous bearing with surface corrugations . Singh , Gupta and Kapur [20] studied the characteristics of corrugated thrust beari ngs. Lin [21] analysed s urface roughness effect on the dynamic stiffness and damping characteristics of compensated hydrostatic thrust bearings . Yacout , Ismaeel and Kassa [22] studied t he combined effects of 2 the centripetal inertia and the surface roughness on the hydrostatic thrust spherical bearings performance . Xuebing et al. [23] investigated the combined effect of the surface soughness and inertia on the performance of high speed hydrostatic t hrust bearing. Walicka et al. [24] analysed various performance characteristics of t hrust bearing with rough surfaces lubricated with non -Newtonian fluids by using Ellis fluid model. Tribologists also worked to increase the stabilizing p roperties of non -Newtonian lubricants by addition of long chain polymer solutions (polyisobutylene and ethylene propylene etc.) as viscosity index improvers [25–27]. The use of additives reduces the sensitivity of the lubricant to the change in the shearing strain rate with temperature due which the lubricant’s characteristics deviate from that of Newtonian fluids [27]. To study the lubrication regimes with such lubricants, m any non -Newtonian models such as couple stress, power law, micropolar and Ca sson models are of common interest . Amongst these models, cubic stress (Rabinowitsch ) fluid model [1–3] is an established model to analyze the non -Newtonian behavior of the fluid. Wada and Hayashi [26] showed that the lubricants blended with viscosity index improver behave like pseudoplastic fluids, which can be analyzed with cubic stress ( Rabinowitsch ) fluid model given by foll owing relation for one-dimensional fluid flow: 3 rz rzu z  (1) where μ̅ is the initial viscosity and κ is the non -linear factor responsible for the non - Newtonian effects of the fluid, which will be referred to as coefficient of pseudoplasticity in this paper. This model is applicable to Newtonia n lubricants ( 0) , dilatant lubricants ( 0) and pseudoplastic lubricants ( 0) . This model is one of the few model s supported by experimental verification [26]. Many researchers used this model for theoretical study of bearing performance w ith non -Newtonian lubricants [28–31]. Singh , Gupta and Kapur [32] analysed the e ffects of inertia in the steady state pressurised flow of a non -Newtonian fluid between two curvilinear surfaces of revolution . Lin [33] studied the squeeze film characteristics between parallel annular disks . Singh , Gupta and Kapur [34–36] investigated the effects of non -Newtonian pseudoplastic lubricants on the various performances of squeeze f ilm, hydrodynamic sliders bea rings. Sharma and Yadav [37] presented the analysis of hydrostatic circular thru st pad bearing . In light of this discussion, it is observed that surface roughness and non -Newtonian lubricants are important considerations for analysis of hydrostatic thrust bearings , and none of the researchers has analysed the combined effects of surfa ce roughness, inertia and non -Newtonian lubricants on hydrostatic thrust bearings using cubic stress fluid model: one of the experimentally verified fluid model. In the present investigation , it is proposed to uncover the combined effects of surface rough ness, non -Newtonian fluids and fluid inertia on the performance of a hydrostatic thrust bearing using cubic stress fluid model . 3 Fig. 1 : Schematic Diagram of Step Thrust Bearing with rough surfaces ANALYSIS A schematic configuration of a hydrostatic t hrust bearing with rough surfaces is shown in Figure 1. It is assumed that the lubricant in the system is incompressible non-Newtonian fluid , the body forces and body couples are absent, and the assumptions of hydrodynamic lubrication are applicable to lubricant film. Following Singh , Gupta and Kapur [1], the expression for pressure gradient is : 5 4 35 3 2 2 3 1 3 3 13 6 20 10 84fpr f r f frQ            (2) where, 322 24612 57 QQfrr   and shh . Here, h is the nominal smooth part of the film th ickness,  is step ratio, sh is the part of film thickness due to asperities on the surface measured from smooth level . In case of radial roughness, ( , )sshh , and in case of transverse or circu mferential roughness, ( , )ssh h r ; where,  is a random variable to 4 characterize some definite arrangement of the surface asperities. The boundary conditions for eq.(2) are : ( ) 1opr and (1) 0p together with the continuity of pressure at the step. Taking the dimensionless probability density function o f the stochastic film thickness ()sfh , Christensen [18]: 322 735;() 32 0 ; elsewheress sc h c h cfh c     (3) and defining the expectancy function ()E as : ( ) ( )ss E f h dh    , (4) stochastic form of eq. ( 2) can be written as : 545 3 2 2 3 3 ( ) 1 3 3 13 6 20 10 84EpE r f r f f frQ              (5) where , c is the half range assumed by of the random film thickness variable  with the standard deviation  . The function ()sfh - defined in eq. ( 3) terminates at 3c . In the present analysis, c will be referred to as roughness parameter. Denoting ()Ep by P and integrating eq. ( 5) with respect to r along the radius of the bearing, following expression is obtained : 5 3 2 2 3 5 4 3 1 3 3 13( ) 16 20 10 84or rP r r f r f f drQf E            (6) where, the evaluation of integral on the right hand side of eq. ( 6) requires the exact expressions for expectation function occurring in the integrand. The dimensionless load carrying capacity is calculated as : 1 1112 0 0( ) 2 ( )d 2 ( )dr o rW rP r dr r rP r r rP r r      (7) PRESSURE DISTRIBUTIO N Case I: Radial Roughness: 5 For the radial roughness, the stochastic film thickness is ( , )shh   and stochastic form of eq. (6) is 5 3 354 32 213( ) 1 ( ) ( ) (6 20 3 13 ( )( ) ( ) ( )10 84)() or rP r r E E f EQ ErEE E f E f drf           (8) where, 2 2 32 727 61()(5) ( )QQEfr E r E   (9) 3 2 3 3 1:3E c h h  (10) 5 4 2 3 3 5 5 5 10:33 9E c h c h h      (11) 7 6 4 3 3 2 5 5 7 7 35 35 7:429 33 3E c h c h c h h        (12) Equations ( 8–12) represent the expression for pressure in the case of radial roughness, which give the pressure in the recess region 1 ()or r r for 1 , and press ure in the land region 1( 1)rr for 1 . Case I I: Circumferential or Azimuthal Roughness: For the circumferential roughness, the stochastic film thickness is ( , )sh h r and stochastic form of eq. ( 6) is 2 3 2 3 3 54 51 ( ) 3 3 ( ) 13 ( )( ) 16 ( ) 20 ( 10 ( ) 84 )() ( ) or rEf E f E f E fP r r r drQ E E E E                  (13) where, 22 7 23 6( ) (527) ( )QQE f ErrE    (14)   3 2 2 2 4 4 4 3 3 3 73 6 5 log 33 250 263hcE c h c h ch c hc h c           (15) 6   5 4 2 2 2 4 4 3 3 3 7 2 2 2353 6 5 log 26 30 32hcE c c h h c h chhc c c h             (16)  32 2 2 3 3 3 5 5 5 3727 2215 log 80 66 30 67 9hcc h c h c h chhc cE ch            (17) Equations, eq. ( 13 –17) give the pressure in the recess region 1 ()or r r for 1 , and land region 1( 1)rr for 1 . Fig. 2 (a) : Variation of film pressure with radius for different values of inertia parameter ( S) and roughness parameter ( c) [Longitudinal Roughness] (b) : Variation of film pressure with radius for different values of inertia parameter ( S) and roug hness parameter ( c) [Circular Roughness] RESULTS AND DISCUSSI ON In order to analyse the effect of roughness on the steady state performance characteristics of externally pressurized thrust bearings lubricated with non-Newtonian (p seudoplastic) lubricants, numerical resu lts for pressure and load capacity have been obtained for the different values of roughness parameter   0 0.005cc and parameter of pseudoplasticity, 2()oP    . For the practical justification, results have been discussed for the experimental values for coefficient of p seudoplasticity, 6 5.65 10 m4/N2 ( 0.3 % 7 additive) and 63.05 10 m4/N2 (1 % additive ) have been taken [26]. The values of operating parameters, namely, f ilm thickness ratio  = 2, 5; parameter of centrifugal inertia S = 0, 1.0, 2.0; supply radius 0.05or , and step radius 10.4r have been taken from experimental results of Dowson [6]. Further, the theoret ical results of pressure for Newtonian and non - Newtonian (pseudoplastic) lubricants have been compared with the experimental results of Dowson [6], and the results for both load capacity and pressure have been compared with the established theoretical results of Singh , Gupta and Kapur [1]. It was observed that pressure and load capacity for smooth surfaces ( 0c ) in the present analysis are same as those obtained by Singh, Gupta and Kapur [1]. Fig. 3 (a) : Variation of film pressure with radius for different values of inertia parameter ( S) and roughness parameter ( c) [Longitudinal Roughness] (b) : Variation of film pressure with radius for different values of inertia parameter ( S) and roughness parameter ( c) [Circular Roughness] Figure 2 shows the variation of film pressure along the radial direction for step ratio 2.18 and different values of roughness parameter ( c), coefficient of pseudoplasticity (  ) and inertia parameter ( s ). To maintain the graphical clarity, the results for longitudinal and circular roughness have been p resented as separate figures. In both the case of longitudinal and circular roughness patterns, the dimensionless pressure for 0c is same as obtained by Singh, Gupta and Kapur [1] for each value of s , which validates the present results for smooth surfaces. It is, further, observed that the longitudinal roughness decreases the dimen sionless values of film pressure , while the circular roughness pattern increases the dimensionless pressure. In case of the longitudinal roughness patterns ( 0.005)c , the pressure is less than that for 0c for Newtoni an as well as non -Newtonian lubricants, and 8 the trend holds for each value of 0,1 and 2s , while the case is reversed in case of circular roughness patterns. Further, in comparison to smooth surface, the theoretical values of pressure obtain ed for rough surfaces with longitudinal roughness are closer to the experimental values . The pressure for longitudinal rough ness ( 0.005c ) and non-Newtonian lubricants 6 5.65 10 are more close to the experimental results [6] than the theoretical results [1]. In figure 3, similar variation of profile of dimensionless pressure has been obtained and compared with established theoretical and experimental results for step ratio 1.54 . This establi shes the sustainability of present results of film pressure for surface roughness. Furthermore, all the figures for pressure also shows that the effects of surface roughness and non -Newtonian lubricants are more significant at higher values of inertia parameter .s Fig. 4 (a) : Variation of load capacity with step parameter ( r1) for different values of roughness parameter ( c), inertia parameter s and coefficient of pseudopasticity  [Longitudinal Rough ness] (b) : Variation of load capacity with step parameter ( r1) for different values of roughness parameter ( c), inertia parameter s and coefficient of pseudopasticity  [Circular Roughness] Figure 4 shows variation of dimensionl ess load capacity with respect to the step position 1r for different values of roughness parameter ( c), coefficient of pseudoplasticity (  ) and inertia parameter ( s ). In case of smoo th surfaces ( 0)c , the load capacity obtained in the present analysis is same as that obtained by Singh, Gupta and Kapur [1] for each value of 0,1 and 2s . The load capacity for surface with longitudinal roughness pattern ( 0.005)c is less than that for smooth surfaces in both the cases of Newtonian and non-Newtonian lubricants, and the trend of variations sustain for each value of s . In case of circular roughness patterns, the results are reversed. Furthermore, it can be clearly observed from the figures that higher is the value of inertia parameter s , more significant are the effects of surface roughness and non -Newtonian lubricants. 9 Figure 5 show the variation of load carrying capacity with roughness parameters ()c for inertia parameter ( 0,1,2)s , Newtonian lubricants ( ( 0) , pseudoplastic lubricants 8( 2.95 10 ) and a typical value of step parameter 10.4r . Results for both the type of longitudinal and circular roughness patterns are plotted simultaneously to observe the influence of surface roughness and its pattern on the load capacity more precisely . It is easy to observe that load carrying capacity decreases with the increase of longitudinal roughness, and increases with the increa se of circular roughness for Newtonian as well as pseudoplastic lubricants at each value of inertia parameter s. It is also clearly observed that surface roughness has more significant influence on load capacity for higher values of inertia parameter s. Fig. 5 : Variation of load capacity with roughness parameter ( c) for differe nt values of inertia parameter s and coefficient of pseudopasticity  [○ – Longitudinal Roughness, ● – Circular Roughness] CONCLUSIONS Combined effects of surface roughness, non -Newtonian pseudoplastic lubricants and lubricant inertia on the steady performance of hydrostatic thrust bearings, neglecting the radial inertia of lubricant and cavitation effects, have been presented. Cubic stress -strain model for non -Newtonian nature of the fluid, Christensen theory for surface roughness and energy integral method to derive pressure gradient were used in the analysis , based on which, following conclusions are drawn – 1. In comparison with smooth surfaces, dimensionless film pressure and load capacity is lower for longitudinal roughness and higher for circular roughness patterns with Newtonian as well as non -Newtonian lubricants . 2. In case of longitudinal roughness patterns, load carrying ca pacity decreases with the increase roughness; and in the case of circular roughness, load capacity increases with the increase roughness. 10 3. In comparison with Newtonian lubricants, dimensionless pressure and load is lower for p seudoplastic lubricants , which is also the established result [1]. 4. Effects of surface roughness and non -Newtonian lubricants become more significant for larg er values of inertia parameter, that is, for larger bearing radius and higher operating speed. Hence, the p resent analysis is expected to be helpful for better bearing designs .