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278 Journal of Marine Science and Technology, Vol. 15, No. 4, pp. 278-286 (2007) EFFECTS OF SURFACE ROUGHNESS ON POROUS INCLINED SLIDER BEARINGS LUBRICATED WITH MICROPOLAR FLUIDS N.B. Naduvinamani* and T.V. Biradar** Key words: surface roughness, micropolar fluids, porous, slider bearings. Reynolds type equation which is applicable to any general surface roughness structure. Chritensen [3] proposed a new stochastic averaging approach for the ABSTRACT study of roughness effects on the hydrodynamic lubri- cation of bearings. Christensen and Tonder [4-6] pre- This paper describes a theoretical analysis of the effect of surface sented a comprehensive general analysis for the two roughness on the hydrodynamic lubrication of one-dimensional po- types of one dimensional surface roughness patterns rous inclined slider bearings with micropolar fluids. To account for more realistic situations of the non-uniform rubbing of the bearing Viz. transverse and longitudinal, based on the general surfaces, it is assumed that, the probability density function for the probability density function and this approach formed random variable characterizing the surface roughness is asymmetrical the basis for the study of surface roughness effects by with non-zero mean. The averaged modified Reynolds type equation several researchers [9, 11, 12, 14]. In all these studies is derived and the closed form expressions for the bearing character- it is assumed that, the probability density function for istics are obtained. The numerical computations of the results show that, the performance of the porous inclined slider bearing is im- the random variable characterizing the surface rough- proved for the micorpolar lubricants as compared to the correspond- ness is symmetric with zero mean. However, in general ing Newtonian lubricants. Further, it is found that, the negatively due to non-uniform rubbing of the surfaces, especially skewed surface roughness improves the porous bearing performance in slider bearings the distribution of surface roughness whereas, the bearing performance suffers due to the presence of may be asymmetrical. In view of this, Andharia et al. positively skewed surface roughness. [2] studied the effect of surface roughness on the perfor- INTRODUCTION mance characteristics of one-dimensional slider bear- ings with an assumption of the probability density func- The study of the effects of surface roughness on tion for the random variable characterizing the surface the hydrodynamic lubrication of various bearing sys- roughness is asymmetrical with a non-zero mean. All tems has been a subject of growing interest. This is these studies are limited to the study of surface rough- mainly because of the reason that, in practice all bearing ness effects on bearing performance with Newtonian surfaces are rough. The study of the effect of surface lubricants. roughness has a greater importance in the study of Most of the modern lubricants are no longer porous bearings as the surface roughness is inherent to Newtonian fluids, since the use of the lubricant addi- the process used in their manufacture. In general, the tives in lubricants has become a common practice in roughness asperity height is of the same order as the order to improve the performance of lubricants. mean separation between the lubricated contacts. In Therefore, several microcontinuum theories [7] have such situations, surface roughness affects the perfor- been proposed to account for the effects of additives. mance of the bearing system. The stochastic study of Eringen [8] micropolar fluid theory is a subclass of Tzeng and Saibel [21] has fascinated several investiga- microfluids that ignores the deformation of the micro- tors in the field of tribology. Patir and Cheng [15, 16] elements and allows for the particle micromotion to take proposed an average flow model for deriving the place. Allen and Kline [1] presented the lubrication theory for micropolar fluids in which the order-of- magnitude arguments are used to simplify the govern- Paper Submitted 08/04/06, Accepted 12/14/06. Author for Correspondence: ing equations to a system of coupled, linear , ordinary N.B. Naduvinamani. E-mail: naduvinamaninb@yahoo.co.in. differential equations. The lubrication theory for *Department of Mathematics, Gulbarga University, Gulbarga-585 106, micropolar fluids and its application to a journal bear- INDIA. **Department of Mathematics, Appa Institute of Engineering and ing is presented by Prakash and Sinha [17]. Shukla and Technology, Gulbarga-585 103, INDIA. Isa [20] derived a generalized Reynolds equation for N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings 279 micropolar lubricants and its application to optimum y one-dimensional slider bearings. Ramanaiah and Dubey [19] presented the analysis for optimum slider profile of a slider bearing lubricated with a micropolar fluid. The micropolarity-roughness interaction in hydrodynamic U lubrication is studied by Praksh et al. [18]. This is based on the Chritensen’s [3] stochastic model for hydrody- namic lubrication of rough surfaces. The porous in- h1 C clined slider bearing lubricated with micropolar fluid is studied by Verma et al. [22] with an assumption of h0 perfectly smooth bearing surfaces. The effect of three- x dimensional irregularities on the hydrodynamic lubri- δ Porous bearing Solid backing cation of journal bearings lubricated with micropolar L lubricants is studied by Lin [13]. In this paper an attempt has been made to study the Fig. 1. Porous plane inclined slider bearing. effect of micropolarity-roughness interaction on the performance of one-dimensional porous inclined slider bearings. It is assumed that, the probability density random variable h s are defined as function for the random variable characterizing the surface roughness is asymmetrical with non-zero mean. α * = E(h s) (2) FORMULATION OF THE PROBLEM σ * = E[(h s – α *) 2] (3) Figure 1 shows the schematic diagram of the po- ε * = E[(h s – α *) 3] (4) rous inclined slider bearing. It consists of two surfaces separated by a lubricant film. The lower surface of the where E is the expectation operator defined by porous bearing is at rest and the upper solid surface is moving in its own plane with a constant velocity U. It ∞ is assumed that the bearing surfaces are rough and E( • ) = ( • ) f (h s)dh s (5) infinitely wide in the z-direction. The lubricant in the –∞ film region as well as in the porous region is assumed to be micropolar fluid. It is to be noted that, the parameters α *, σ * and ε * are To mathematically model the surface roughness, independent of x, the mean α* and the parameter ε * can expression for the film thickness is considered to be assume both positive and negative values however σ * consisting of two parts. can always assume positive values. The field equations for micropolar fluids proposed H = h(x) + h s (1) by Eringen [8] simplify considerably under the usual assumptions of hydrodynamic lubrication i.e. laminar, where incompressible fluid with negligible body forces and negligible inertia forces. The resulting equations under h1 – h0 steady state conditions given by [22] are; h(x) = h 1 – x Conservation of mass: L ∂u ∂v + =0 (6) ∂x ∂y is the mean film thickness, h s is a randomly varying quantity measured from the mean level and thus charac- Conservation of momentum: terizes the surface roughness and L is the length of the bearing. The stochastic part h s is assumed to have the ∂ 2u ∂ν ∂p probability density function f(h s) defined over the do- (µ + K) +K – =0 (7) ∂y 2 ∂y ∂x main –C ≤ h s ≤ C where C is the maximum deviation from the mean level. For a continuous non-broken fluid ∂p film thickness it is assumed that C < < h(x). =0 (8) The mean α * , the standard deviation σ * and the ∂y parameter ε *, which is the measure of symmetry of the Conservation of angular momentum: 280 Journal of Marine Science and Technology, Vol. 15, No. 4 (2007) ∂ 2ν ∂u h Mh γ –K – 2Kν = 0 (9) + 1 – tanh2 Mα * M3 ∂y 2 ∂y h0 h0 – A h0 3 where γ and K are the viscosity coefficients for 2 α* Mh Mh micropolar fluids. + tanh + M 1 – tanh2 B The flow of micropolar lubricants in the porous matrix h0 h0 h0 is governed by the modified Darcy’s law h3 Mh –k + 2M2(1 + λ) tanh q= ∇p * (10) 3 h0 h0 µ+K where q = (u*, v*) is the modified Darcy velocity vector, h3 Mh Mα * M3 + 1 – tanh – A k is the permeability of the porous matrix and p * is the 3 h0 h0 h0 3 pressure in the porous matrix. The modified Reynolds equation for smooth po- rous inclined slider bearing lubricated with micropolar Mh 3h 2α * Mh 3h 2 Mh fluid was obtained by [22] in the form + tanh A + 3 tanh + 2 1 – tanh MB h0 h0 h0 h0 h0 ∂ kδ ∂p U ∂H f (H, M) + = (11) 3hB Mh 3h Mh ∂x µ + K ∂x 2 ∂x + tanh + 1 – tanh2 MA – 3λ MB h0 h0 h0 h0 where h3 2 3 – 3λ H M + 3λ H tanh(MH ) + 2H M2(1 + λ)tanh(MH ) 1 f (H, M) = 0 * µ(2 + λ) 12M 2(1 + λ)tanh(M H ) Mh Mh Mα * M3 12M2(1 + λ ) tanh + 1 – tanh2 – A h0 h0 h0 3 (12) where (14) 1 µ H mh K(2µ + K) 2 with λ = , H = , M = 0, m = and δ is the K h0 2 γ (µ + K) 3 2 2 2 ε* α* σ* α* α* σ* porous layer thickness. A= + +3 and B = + h3 0 h3 0 h2 0 h0 h0 h2 0 Multiplying both sides of Eq.(11) by h s and inte- grating with respect to h s over the interval –C to C and using Eqs. (2)-(4), gives the averaged Reynolds type The relevant boundary conditions for the pressure field equation in the form are – p=0 at x = 0, L (15) ∂ * * * kδ ∂p U ∂h G(h, M, α , σ , ε ) + = (13) ∂x µ + K ∂x 2 ∂x Introducing the non-dimensional scheme – where p is the expected value of p and x h hs p h20 kδ x = , h = , hs = , P = , ψ= 3 L h0 h0 µUL h 0 h3 0 h Mh K α* σ* ε* G(h, M, α *, σ *, ε *) = 3λ tanh λ= , α= , σ= 2 , ε= 3 µ(2 + λ) h0 h0 µ h0 (16) h h 0 0 N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings 281 into the Eqs. (13) and (15) gives the non-dimensional –1 ψ averaged Reynolds type equation and boundary condi- + D G(h , M, α, σ, ε) + dx (22) tions in the form 1+λ The component of stress tensor required for computing ∂ ψ ∂P 1 ∂h G(h , M, α, σ, ε) + = (17) frictional force is ∂x 1 + λ ∂x 2 ∂x – = 0, 1 1 ∂u P=0 at x (18) τyx = (2µ + K) – Kv (23) 2 ∂y BEARING CHARACTERISTICS The frictional force F per unit width on the sliding surface at y = H is defined as The solution of Eq. (17) subject to boundary con- L ditions (18) is obtained in the form F= (τyx)y = H dx (24) P = I 1 + DI 2 (19) 0 where where x –1 K ∂u h ψ (τyx)y = H = µ + (25) I1 = G(h , M, α, σ, ε) + dx , 2 ∂y 2 1+λ y=H 0 since v = 0 at y = H x –1 ψ Using the solution of Eq. (7) in Eq. (25) and then I2 = G(h , M, α, σ, ε) + dx , in Eq. (24) and the use of non-dimensional scheme 1+λ 0 gives the non-dimensional mean frictional force in the I3 form D=– , I4 1 1 h + α dP M(1 + λ)(2 + λ) –1 F = + dx h ψ 2 dx 4Mh (h + α)(1 + λ) + λg I3 = G(h , M, α, σ, ε) + dx and 0 2 1+λ 0 (26) 1 –1 ψ where I4 = G(h , M, α, σ, ε) + dx . (20) 1+λ 0 M3 The non-dimensional mean load carrying capacity is g = tanh(Mh ) + 1 – tanh2(Mh ) Mα – (ε + α 3 + α 2σ) 3 obtained in the form [22]; 1 1 dP (27) W = Pdx = – x dx (21) dx 0 0 The non-dimensional coefficient of friction is then ob- tained by Use of Eq. (19) in (21) gives F Cf = (28) 1 –1 W h ψ W = – x G(h , M, α, σ, ε) + 2 1+λ 0 RESULTS AND DISCUSSION The Effect of surface roughness and micropolar 282 Journal of Marine Science and Technology, Vol. 15, No. 4 (2007) lubricants on the performance characteristics of the parameter ψ . The integrations I1, I2 and I 3 involved in porous inclined slider bearings is studied through the the calculation of non-dimensional pressure P given in dimensionless parameters α , ε , σ , M, ψ and λ . As the Eq. (17) are calculated numerically [10]. It is observed roughness parameters tends to zero the results obtained that P increases for increasing values of M and for in this paper reduce to the smooth case studied by decreasing value of ψ . Further, it is observed that the Verma et al. [22]. For the numerical computations of point of maximum pressure is a function of the perme- the slider characteristics, the following set of values are ability parameter ψ . used for various non-dimensional parameters α = -0.1 – The effect of roughness parameter α , ε and σ on 0.1; ε = -0.1 – 0.1; σ = 0.0 – 0.4; λ = 0.0 – 0.4; M = 4, – the variations of P with x is shown in the Figures 3, 4 6, 8, 10; and ψ = 0.1, 0.2. The numerical values for the and 5 respectively for two values of the parameter M. It roughness parameters α , ε, σ are also so chosen that the is observed from Figures 3 and 4 that P increases for corresponding film shapes are feasible. negatively increasing values of α and ε whereas P decreases for positively increasing values of α and ε . 1. Pressure The increasing values of σ decreases the fluid film pressure P (Figure 5). The effect of micropolar lubricants on the varia- tion of the non-dimensional pressure P with – is de- x 2. Load carrying capacity picted in Figure 2 for two values of the permeability Figure 6 shows the variation of non-dimensional 0.5 ψ = 0.1 ψ = 0.2 M=4 0.5 0.4 M = 6 M=8 M = 10 0.4 0.3 P 0.3 0.2 0.2 M = 4 M = 10 0.1 P ε = -0.1 ε = -0.05 0.0 0.1 ε = 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ε = 0.05 – x ε = 0.1 0.0 – Fig. 2. Variation of non-dimensional pressure P with x for different 0.2 0.4 0.6 0.8 1.0 values of M with α = -0.1, σ = 0.1, ε = -0.1 and λ = 2. – x – Fig. 4. Variation of non-dimensional pressure P with x for different values of ε with α = -0.1, σ = 0.1, ψ = 0.1 and λ = 2. 0.5 M = 4 M = 10 α = -0.1 α = -0.05 0.5 0.4 α = 0.0 α = 0.05 α = 0.1 0.4 0.3 P 0.3 0.2 M = 4 M = 10 P 0.2 σ = 0.0 σ = 0.1 0.1 σ = 0.2 0.1 σ = 0.3 σ = 0.4 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 – – x x – Fig. 3. Variation of non-dimensional pressure P with x for different – Fig. 5. Variation of non-dimensional pressure P with x for different values of α with σ = 0.1, ε = -0.1, ψ = 0.1 and λ = 2. values of σ with α = -0.1, ε = -0.1, ψ = 0.1 and λ = 2. N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings 283 load carrying capacity W with λ for different values of where ψ = 0.2. – M. The numerical value of λ = 0 corresponds to the Figures 11, 12 and 13 shows the variation of F Newtonian case. It is observed that W increases for with λ for different values of α, ε and σ respectively for increasing values of λ and M. The effect of roughness two values of M. It is observed that negatively skewed – parameter α , ε and σ on the variation of W with λ is surface roughness pattern decreases F whereas posi- depicted in the Figures 7, 8 and 9 respectively for two tively skewed surface roughness increases the frictional values of M. It is observed that, negatively skewed force on the sliding surface. surface roughness increases W whereas positively skewed surface roughness decreases W . Further, the 4. Coefficient of friction significant increase in W is observed for larger values of λ as compared to the Newtonian case ( λ = 0). The variation of non-dimensional coefficient of friction C f with the parameter λ for different values of 3. Frictional force M is shown in the Figure 14 for two values of permeabil- ity parameter ψ . It is observed that C f decreases Figure 10 shows the variation of non-dimensional rapidly for increasing values of λ as compared to the – frictional force F with λ for different values of M. It is Newtonian case ( λ = 0). However, the marginal in- – observed that, for ψ = 0.1 F decreases up to the value of crease in C f is observed for increasing values of M. – λ = 0.5 and then F increases for further increasing The effect of roughness parameters α , ε and σ on – values of λ . However, F increases for all values of λ M = 10 M = 4 ψ = 0.1 ψ = 0.2 1.4 ε = -0.1 1.4 M = 4 M = 10 ε = -0.05 M=6 1.2 ε = 0.0 1.2 M = 8 ψ = 0.1 ε = 0.05 M=4 M = 10 ε = 0.1 ψ = 0.2 1.0 1.0 W 0.8 W 0.8 0.6 0.6 0.4 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0 1 2 3 4 λ λ Fig. 8. Variation of non-dimensional load W with λ for different Fig. 6. Variation of non-dimensional load W with λ for different values of ε with α = -0.1, σ = 0.1 and ψ = 0.1. values of M with α = -0.1, σ = 0.1 and ε = -0.1. M = 4 M = 10 1.4 σ = 0.0 1.4 α = -0.1 Μ = 4 Μ = 10 σ = 0.1 σ = 0.2 α = -0.05 1.2 σ = 0.3 1.2 α = 0.0 σ = 0.4 α = 0.05 1.0 α = 0.1 1.0 0.8 W 0.8 W 0.6 0.6 0.4 0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 λ λ Fig. 7. Variation of non-dimensional load W with λ for different Fig. 9. Variation of non-dimensional load W with λ for different values of α with σ = 0.1, ε = -0.1 and ψ = 0.1. values of σ with α = -0.1, ε = -0.1 and ψ = 0.1. 284 Journal of Marine Science and Technology, Vol. 15, No. 4 (2007) 1.05 1.00 ψ = 0.1 ψ = 0.1 M = 4 M = 10 0.95 M = 4 1.00 σ = 0.0 M=6 0.95 σ = 0.1 0.90 M = 8 σ = 0.2 M = 10 0.90 σ = 0.3 0.85 σ = 0.4 0.80 0.85 0.75 0.80 F 0.75 M = 10 F 0.70 0.65 0.70 0.60 0.65 M=4 0.55 0.60 0.50 0.55 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 λ λ – – Fig. 10. Variation of non-dimensional frictional force F with λ for Fig. 13. Variation of non-dimensional frictional force F with λ for different values of M with α = -0.1, σ = 0.1 and ε = -0.1. different values of σ with α = -0.1, ε = -0.1 and ψ = 0.1. 1.0 1.10 M = 4 M = 10 ψ = 0.1 ψ = 0.2 1.05 α = -0.1 0.9 M=4 1.00 α = -0.05 0.8 M=6 M=8 0.95 α = 0.0 M = 10 α = 0.05 0.7 0.90 α = 0.1 0.85 0.6 ψ = 0.2 ψ = 0.1 F 0.80 C 0.5 0.75 0.4 0.70 0.65 0.3 0.60 0.2 0.55 0.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 . 1.0 1.5 2.0 2.5 3.0 3.5 4.0 λ λ – Fig. 11. Variation of non-dimensional frictional force F with λ for Fig. 14. Variation of non-dimensional Coefficient of friction C f with λ different values of α with σ = 0.1, ε = -0.1 and ψ = 0.1. for different values of M with α = -0.1, σ = 0.1 and ε = -0.1 1.0 1.00 M = 4 M = 10 M = 4 M = 10 ε = -0.1 0.9 α = -0.1 0.95 ε = -0.05 α = -0.05 0.90 ε = 0.0 0.8 α = 0.0 ε = 0.05 α = 0.05 0.85 ε = 0.1 0.7 α = 0.1 0.80 0.6 F 0.75 M = 10 C 0.5 0.70 M=4 0.4 0.65 0.60 0.3 0.55 0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.1 λ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 λ – Fig. 12. Variation of non-dimensional frictional force F with λ for Fig. 15. Variation of non-dimensional Coefficient of friction C f with λ different values of ε with α = -0.1, σ = 0.1 and ψ = 0.1. for different values of α with σ = 0.1, ε = -0.1 and ψ = 0.1. N.B. Naduvinamani & T.V. Biradar: Effects of Surface Roughness on Porous Inclined Slider Bearings 285 1.0 0.9 M = 4 M = 10 M = 4 M = 10 ε = -0.1 0.9 σ = 0.0 0.8 ε = -0.05 0.8 σ = 0.1 ε = 0.0 σ = 0.2 0.7 ε = 0.05 0.7 σ = 0.3 ε = 0.1 σ = 0.4 0.6 0.6 0.5 C 0.5 C 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 λ λ Fig. 16. Variation of non-dimensional Coefficient of friction C f with λ Fig. 17. Variation of non-dimensional Coefficient of friction C f with λ for different values of ε with α = -0.1, σ = 0.1 and ψ = 0.1. for different values of σ with α = -0.1, ε = -0.1 and ψ = 0.1. the variation of C f with λ is depicted in Figures 15, 16 Sharnbasawappa Appa, President , Sharnbasveshwar and 17. It is observed that C f decreases for negatively Vidya Vardhak Sangha, Gulbarga for his encour- skewed surface roughness pattern whereas it increases agements. This work is supported by the UGC New for positively skewed surfaces roughness pattern. Delhi under the Project No.F-31-84/2005(SR). Further, the increasing values of σ increases the coeffi- cient of friction C f marginally. These increase/de- NOMENCLATURE crease in C f are more pronounced for larger values of λ. C maximum asperity height from the mean level CONCLUSIONS Cf non-dimensional frictional coefficient, The performance characteristics of the rough po- rous inclined slider bearings lubricated with micropolar Cf = F W fluids is analysed on the basis of Eringen [8] constitu- tive equations for micropolar fluids and the stochastic E expectancy operator defined by equation (5) random variable to represent the surface roughness of F frictional force the bearing with non-zero mean variance and skewness. – Fh o On the basis of the numerical computations performed, F non-dimensional frictional force, F = µUL the following conclusions can be drawn. H fluid film thickness, H = h + h s 1. The significant increase in the load carrying capacity and decrease in the coefficient of friction can be h1 – h0 attained by the use of micropolar lubricants. h mean film thickness, h(x) = h 1 – x L 2. The porous facing on the slider bearing affects its performance. – 3. The presence of negatively skewed surface roughness h non-dimensional mean film thickness, h = h h o pattern on the bearing surface improves its – performance. H non-dimensional fluid film thickness, H = H h o 4. The positively skewed surface roughness pattern on the porous slider bearing affects its performance. ho outlet film thickness 5. The above effects are more accentuated for larger h1 inlet film thickness values of the parameter λ . hs random variable K viscosity coefficient for micropolar fluid. ACKNOWLEDGEMENTS k permeability of the porous matrix. L bearing length The authors are grateful to the referees for their M non-dimensional micropolar parameter, valuable comments on the earlier draft of the paper. One of the authors (TVB) is thankful to Poojya Dr. M = mh o 2 286 Journal of Marine Science and Technology, Vol. 15, No. 4 (2007) 1 8. Eringen, A.C., “Theory of Micropolar Fluids,” Journal 2 K(2µ + K) m micropolar parameter, m = of Mathematics and Mechanics, Vol. 16, No. 1, pp. 1-18 γ (µ + K) (1966). p pressure in the film region. 9. Gururajan, K. and Prakash, J., “Effect of Velocity Slip in a Narrow Rough Porous Journal Bearing,” Journal of – p expected value of p Engineering Tribology, Vol. 217, pp. 59-69 (2003). 10. Jain, M.K., Iyengar, S.R.K., and Jain, R.K., Numerical 2 P non-dimensional pressure, P = ph o µUL Methods for Scientific and Engineering Computations, New Age International Publishers, New Delhi, ND U velocity of the slider (2002). 11. Letalleur, N., Plourbouc, F., and Prat, M., “Average u, v fluid velocities in the x and y directions Flow Model of Rough Surface Lubrication,” ASME u *, v * modified Darcy velocity components in the x Journal of Tribology, Vol. 124, pp. 539-545 (2002). and y directions 12. Lin, J.R., Hsu, C.H., and Lai, C., “Surface Roughness W non-dimensional load carrying capacity Effects on the Oscillating Squeeze Film Behavior of x, y, z Cartesian coordinates Long Partial Journal Bearings,” Computers & Structures, – Vol. 80, pp. 297-303 (2002). x non-dimensional form of x, x = x L 13. Lin, T.R., “Hydrodynamic Lubrication of Journal Bear- ings Including Micropolar Lubricants and Three-Di- α* mean defined by Eq. (2) mensional Irregularities,” WEAR, Vol. 192, pp. 21-28 ε* roughness parameter defined by Eq. (4) (1996). γ viscosity coefficient for micropolar fluid 14. Naduvinamani, N.B. and Gurubasavaraj, G., “Surface µ viscosity coefficient Roughness Effects on Squeeze Films in Curved Circular v angular velocity Plates,” Industrial Lubrication and Tribology, Vol. 56, σ* standard deviation defined by Eq. (3) No. 6, pp. 346-352 (2004). 15. Patir, N. and Cheng, H., “An Average Flow Model for kδ Determining Effects of Three-Dimensional Roughness ψ permeability parameter, ψ = h3 0 on Partial Hydrodynamic Lubrication,” ASME Journal of Lubrication Technolog, Vol. 100, pp. 12-17 (1978). REFERENCES 16. Patir, N. and Cheng, H., “Application of Average Flow Model to Lubrication Between Rough Sliding 1. Allen, S.J. and Kline, K.A., “Lubrication Theory for Surfaces,” ASME Journal of Lubrication Technolog, Micropolar Fluids,” Journal of Applied Mechanics, Vol. Vol. 101, pp. 220-230 (1979). 38, pp. 646-650 (1971). 17. Prakash, J. and Sinha, P., “Lubrication Theory of 2. Andharia, P.I., Gupta, J.L., and Deheri, G.M., “Effect of Micropolar Fluids and Its Application to a Journal Surface Roughness on Hydrodynamic Lubrication of Bearing,” International Journal of Engineering Science, Slider Bearings,” Tribology Transaction, Vol. 44, No. 2, Vol. 13, pp. 217-232 (1975). pp. 291-297 (2001). 18. Prakash, J., Tonder, K., and Christensen, H., “Micro- 3. Christensen, H., “Stochastic Models for Hydrodynamic polarity Roughness Interaction in Hydrodynamic Lubrication of Rough Surfaces,” International Journal Lubrication,” Transaction of ASME, Vol. 102, pp. of Mechanical Engineering, Vol. 184, pp. 1013-1026 368-372 (1980). (1970). 19. Ramanaiah, G. and Dubey, J.N., “Optimum Slider Pro- 4. Christensen, H. and Tonder, K., “Tribology of Rough file of a Slider Bearing Lubricated with a Micropolar Surfaces; Stochastic Models of Hydrodynamic Lubri- Fluid,” WEAR, Vol. 42, pp. 1-7 (1977). cation,” SINTEF Report, No. 10, pp. 69-18 (1969). 20. Shukla, J.B. and Isa, M., “Generalized Reynolds Type 5. Christensen, H. and Tonder, K., “The Hydrodynamic Equation for Micropolar Lubricants and Its Application Lubrication of Rough Bearing Surfaces of Finite to Optimum One-Dimensional Slider Bearing Effects of Width,” ASME Journal of Lubrication Technolog, Vol. Solid Particle Additives in Solution,” Journal of 93, pp. 324-330 (1971). Mechanical Engineering Science, Vol. 17, pp. 280-284 6. Christensen, H. and Tonder, K., “Waviness and Rough- (1975). ness in Hydrodynamic Lubrication,” Proceedings of 21. Tzen, S.T. and Saibel, E., “Surface Roughness Effects on Institution of Mechanical Engineers, Vol. 186, No. 72, Slider Bearing Lubrication,” ASLE Transactions, Vol. pp. 807-812 (1972). 10, pp. 334-338 (1967). 7. Eringen, A.C., “Simple Microfluids,” International 22. Verma, P.D.S., Agrawal, V.K., and Bhatt, S.B., “Porous Journal of Engineering Science, Vol. 2, pp. 205-217 Inclined Slider Bearing Lubricated with Micropolar (1964). Fluid,” WEAR, Vol. 53, pp. 101-106 (1979).

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