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A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing

A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing Hindawi Advances in Tribology Volume 2018, Article ID 3981087, 7 pages https://doi.org/10.1155/2018/3981087 Research Article A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing 1 2 3 4 Jatinkumar V. Adeshara, M. B. Prajapati, G. M. Deheri , and R. M. Patel Research Scholar, Department of Mathematics, H. N. G. University, Patan, 384 265, Gujarat State, India Head, Department of Mathematics, HNG University, Patan 384 265, Gujarat, India Department of Mathematics, SP University, Vallabh Vidyanagar 388 120, Gujarat, India Department of Mathematics, Gujarat Arts and Science College, Ahmedabad 380 006 Gujarat, India Correspondence should be addressed to R. M. Patel; rmpatel2711@gmail.com Received 13 May 2018; Accepted 10 July 2018; Published 2 September 2018 Academic Editor: Huseyin C¸imenogl ˇ u Copyright © 2018 Jatinkumar V. Adeshara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article discusses the effect of longitudinal roughness on the performance of hydromagnetic squeeze film in circular step bearing. To characterize the random roughness of the bearing surfaces the stochastic model of Christensen and Tonder has been employed. eTh stochastically averaged Reynolds’ type equation is solved using suitable boundary conditions to obtain the pressure distribution and then the load bearing capacity is computed. The results are presented in graphical form. eTh graphical results presented here establish that the hydromagnetic lubrication oeff rs significant help to the longitudinal roughness pattern to enhance the performance of the bearing system. Of course, conductivities of the plates, standard deviation, and the supply pressure contribute towards reducing the negative effect induced by variance (+ve) and skewness (+ve). 1. Introduction of the discussions have considered the stochastic averaging method of Christensen and Tonder [5–7] who presented a It is documented that gears, braking units, hydraulic dampers, stochastic model to evaluate the effect of rough surfaces. skeletal bearings, and synovial joints make use of squeeze (Ting [8], Prakash and Tiwari [9], Prajapati [10], and And- film mechanism. Generally an electrically conducting u fl id haria et al. [11]). with high thermal and electrical conductivity is applied as The hydromagnetic squeeze lm fi s between two conduct- a lubricant for squeeze lm fi to work under such extreme ing rough circular plates are studied by Vadher et al. [12]. circumstances. Also, a use of external magnetic eld fi then It was investigated that the negative effect of roughness got advances the performance of lubrication. minimized up to some extent by applying magnetization. The performance of an oil lubricated circular step bearing Patel and Deheri [13] evaluated the performance of ferrou fl id is analyzed by Majumdar [1]; this study underlines the squeeze lm fi in rotating curved circular plates by using importance of radii ratio. Lin [2] discussed the couple Shliomis model. Here, the outcomes of this article suggested stress effect on the performance of an externally pressurized that Shliomis model based ferrofluid lubrication was rela- circular step bearing. It was established that the couple tively better than the model of Neuringer−Rosensweig [4]. stress uid fl modified the performance of the bearing system. Patel et al. [14] extended the study of Deheri et al. [3] to study Deheri et al. [3] reformed and developed the study of the effect of transverse surface roughness. It was indicated Majumdar [1] to use the magnetization effect by considering that the bearing performance was adversely aeff cted by the a magnetic uid fl as the lubricant, the flow being regulated by Neuringer−Rosensweig’s model [4]. roughness of the bearing surfaces. But the situation was found It is well known that the effect of surface roughness is to be a little enhanced in case of variance (-ve) was con- quite important in different types of bearing systems. Most sidered. A modified lubrication equation for hydromagnetic 2 Advances in Tribology " " 0 0 M ,B 1 1 s, H M ,B 0 0 dr Figure 1: Configuration of bearing system. non-Newtonian cylindrical squeeze lm fi s was presented by (2) The bearing has very low rotational velocity and its Lin et al. [15]. The hydromagnetic non-Newtonian effects effect is neglected for the pressure generation. oer ff ed quite better performance for circular squeeze lm fi s As shown in Figure 1 a thrust load w is applied and the bearing in comparison with the case of a nonconducting Newtonian supports the load without metal to metal contact. The load w lubricant. is supported by the u fl id within the pocket and land. The uid fl Patel and Deheri [16] studied the magnetic u fl id lubrica- escapes in the radial direction through the restrictions by a tion of a squeeze lfi m in transversely rough porous circular land or sill around the recess. plates with concentric circular pocket. Longitudinal rough- In view of the discussions of Christensen and Tonder [5– ness effect has been a matter of discussions in Andharia and 7] the surfaces of bearing are taken to be longitudinally rough. Deheri [17], Andharia and Deheri [18], Andharia and Deheri The thickness h(x) of the lubricant lm fi is given by [19], and Shimpi and Deheri [20]. Recently, Lin [21] examined the ferrou fl id lubrication of a longitudinally rough journal h(x)= h(x)+ h (1) bearing. where h(x) is the mean lm fi thickness and h is the deviation Patel et al. [22] discussed the hydromagnetic lubrication s from the mean lm fi thickness characterizing the random of a rough porous circular step bearing. Hence, it has been roughness surface. h is considered in nature and governed by sought to analyze the effect of longitudinal roughness pattern s the probability density function f(h),− c≤ f(h)≤ c, where on behavior of a hydromagnetic squeeze lm fi in circular step s s c is the maximum deviation from the mean lm fi thickness. bearings. The mean 𝛼 and standard deviation 𝜎 and the measure of symmetry 𝜀 of the random variable h are defined by the 2. Analysis relationships. The configuration of the bearing system is presented in 𝛼= E(h) Figure 1. (2) For the computation of different performance character- 𝜎 = E[(h −𝛼) ] istics, generally the following assumptions are considered: and (1) The recess is deep enough for the pressure in it to be (3) uniform. 𝜀= E[(h −𝛼) ] s Advances in Tribology 3 where E denotes the expected value defined by The concerned Reynolds’ type equation gives the pressure induced ow fl for a circular step bearing as (Majumdar (1985), Patel, Deheri, and Vadher (2015)) E(R)=∫ Rf(h) dh (4) s s −c 2𝜋 r(dp/dr)[(2/ M m(h))[M/2− tanh(M/2)]][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] 0 1 0 1 (5) Q=− 12𝜇 r= r; where p= p; −3 −1 −2 2 2 m(h)= h [1−3𝛼 h +3h (𝜎 +𝛼 ) (7) (6) −3 2 3 − h (𝜀+3𝜎 𝛼+𝛼 )] the governing equation for the film pressure p is given by Using Reynolds’ boundary conditions ln(r/r ) p= p (8) ln(r/r ) i o r= r ; p=0; where in p = ln( ) (9) 𝜋[(2/ M m(h)) M/2− tanh(M/2)][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] r [ ] 0 1 0 1 i Introduction of the dimensionless terms 6 ln(1/k) P = 𝜋((2/ M)/ M∗ h ) M/2− tanh M/2 [(𝜙 +𝜙 +1)/(𝜙 +𝜙 + tanh M/2 / M/2)] ( ( )) [ ( )] ( ( )) ( ) 0 1 0 1 3 ∗ ∗2 ∗2 ∗ ∗2 ∗ ∗3 M∗(h)= m(h) h =(1−3𝛼 +3(𝛼 +𝜎 )− (𝜀 +3𝜎 𝛼 +𝛼 ) 𝛼 =( ) (10) 𝜎 =( ) 𝜀 =( ) k=( ) paves the way for the expression of pressure distribution in 3. Results and Discussions nondimensional form as The representation for dimensionless pressure prole fi is given ln(r/r ) in (11) while the nondimensional load bearing capacity is ∗ o P= P (11) governed by (12). From these equations it is clearly observed ln(r/r ) i o that the load carrying capacity The load bearing capacity w is calculated by integrating the pressure which takes the dimensionless form W∝ P (13) ∗ 2 P ln(1− k ) (12) W= and ln(1/k) 4 Advances in Tribology 5.59 4.34 3.09 1.84 0.59 4.00 6.00 8.00 10.00 12.00 0+1= 0 0+1= 3 0+1= 1 0+1= 4 0+1= 2 Figure 2: Variation of load carrying capacity with respect to M and 𝜙 +𝜙 . 0 1 4.99 3.99 2.99 1.99 0.99 4.00 6.00 8.00 10.00 12.00 ∗=0 ∗=0.15 ∗=0.05 ∗=0.2 ∗=0.1 Figure 3: Distribution of load bearing capacity with respect to M and 𝜎 . p = (14) ((2/M)/m(h))[M/2− tanh(M/2)][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] 0 1 0 1 This suggests that for a constant ow fl rate the load increases load with respect to𝜙 +𝜙 for different values of standard 0 1 as stochastically averaged squeeze lm fi thickness decreases. deviation, variance, skewness, and radii ratio, respectively. Therefore, the bearing turns out to be self-compensating Here, also the load carrying capacity increases with an provided the flow rate is considered as constant. It is depicted increase in conductivity parameter. Figures 7–10 also tells that from (11) and (12) that the effect of conductivity parameters the load increases sharply up to some extent (𝜙 +𝜙 ≅1.32) 0 1 on the distribution of pressure and load bearing capacity is and then this rate of increase gets slower. determined by It is seen from Figures 11–13 that the standard deviation associated with longitudinal roughness causes increased load 𝜙 +𝜙 + tanh M/2 / M/2 ( ( )) ( ) 0 1 (15) bearing capacity with respect to various parameters such as 𝜙 +𝜙 +1 0 1 variance, skewness, and radii ratio. Besides, variation of load carrying capacity for different values of variance is described which turns to by Figures 14 and 15. From these figures it is noticed that 𝜙 +𝜙 0 1 the load decreases with increase in positive variance while (16) 𝜙 +𝜙 +1 0 1 the variance (− ve) increases it. The trends of load bearing capacity with respect to skewness are similar to that of for large values of M, because tanh(M) ≅ 1and (2/M) ≅ variance which is observed from Figure 16. 0. Further, it is noticed that the pressure and load carrying Form Figures 5 and 9 it is clearly observed that the effect capacity increases with increases in𝜙 +𝜙 because both the 0 1 functions are increasing functions of𝜙 +𝜙 . of𝜀 on the variation of load carrying capacity with respect 0 1 It is clearly depicted from Figures 2–6 that the hydro- to magnetization and conductivity, respectively, remains negligible while this effect of 𝜀 with regards to variance is magnetization parameter induces a sharp increase in the load bearing capacity while Figures 7–10 gives the profile of registered to be nominal as suggested by Figure 14. LOAD LOAD Advances in Tribology 5 5.66 2.06 4.41 1.66 1.26 3.16 0.86 1.91 0+1 0.46 0.66 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=−0.10 ∗=0.05 ∗=−0.05 ∗=0.10 ∗=−0.05 ∗=0.10 ∗=0 ∗=0 Figure 8: Variation of load carrying capacity with respect to 𝜙 +𝜙 Figure 4: Profile of load with respect to M and 𝛼 . 0 1 and𝛼 . 5.00 1.70 4.00 1.45 3.00 1.20 2.00 0.95 0+1 1.00 0.70 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=−0.10 ∗=0.05 ∗=−0.05 ∗=0.10 ∗=−0.05 ∗=0.10 ∗=0 ∗=0 Figure 5: Variation of load carrying capacity with respect to M and Figure 9: Distribution of load bearing capacity with respect to 𝜙 + 𝜀 . 𝜙 and𝜀 . 5.63 2.04 4.38 1.64 3.13 1.24 1.88 0.84 0+1 0.63 0.44 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 k=0.35 k=0.65 k=0.35 k=0.65 k=0.45 k=0.75 k=0.45 k=0.75 k=0.55 k=0.55 Figure 6: Distribution of load bearing capacity with respect to M Figure 10: Profile of load with respect to 𝜙 +𝜙 and k. 0 1 and k. 2.06 2.10 1.81 1.75 1.56 1.40 ∗ 1.31 1.05 0+1 1.06 0.70 0.00 0.05 0.10 0.15 0.20 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=0 ∗=0.15 ∗=−0.05 ∗=0.10 ∗=0.05 ∗=0.20 ∗=0 ∗=0.10 Figure 11: Variation of load carrying capacity with respect to 𝜎 and ∗ ∗ Figure 7: Profile of load with respect to 𝜙 +𝜙 and𝜎 . 𝛼 . 0 1 LOAD LOAD LOAD LOAD LOAD LOAD LOAD LOAD 6 Advances in Tribology 2.02 1.91 1.82 1.81 1.62 1.42 1.71 ∗ 1.22 ∗ 1.61 1.02 0.00 0.05 0.10 0.15 0.20 −0.10 −0.05 0.00 0.05 0.10 ∗=−0.10 ∗=0.05 k=0.35 k=0.65 ∗=−0.05 ∗=0.10 k=0.45 k=0.75 ∗=0 k=0.55 ∗ ∗ Figure 12: Distribution of load bearing capacity with respect to 𝜎 Figure 16: Profile of load with respect to 𝜀 and k. and𝜀 . This means that this type of bearing system the suitable 2.27 combination of negatively skewed roughness and negative 2.02 variance maygo a long wayfor better performance of the 1.77 bearing system. 1.52 It is seen that for small as well as large values of M 1.27 the performance of the bearing suffers when the plates are ∗ 1.02 considered to be electrically conducting, in matching with 0.00 0.05 0.10 0.15 0.20 the hydromagnetic case, when the plates are taken to be nonconducting. Further, this can physically be clarified by k=0.35 k=0.65 fringing phenomena which happens when the plates are k=0.45 k=0.75 electrically conducting. Finally, the load is acquired to be k=0.55 more in comparison with the Neuringer Rosensweig model Figure 13: Profile of load with respect to 𝜎 and k. based ferrou fl id squeeze lm. fi 1.87 4. Conclusions 1.67 The trio of hydromagnetization, supply pressure, and stan- dard deviation counters fruitfully the negative effect induced 1.47 by positively skewed roughness and variance (+ve) in the 1.27 case of negatively skewed roughness when variance comes ∗ out to be negative. Of course, in bettering the bearing 1.07 performancethe plateconductivities and radiiratio may play −0.10 −0.05 0.00 0.05 0.10 an important role. Needless to say, the longitudinal roughness ∗=−0.10 ∗=0.05 parameter behaves in a little better way as compared to ∗=−0.05 ∗=0.10 the transverse roughness. Therefore, it becomes very much ∗=0 essential that the longitudinal roughness aspect must be Figure 14: Variation of load carrying capacity with respect to 𝛼 and given due consideration from longevity point of view while 𝜀 . designing this type of bearing system. 2.28 Nomenclature 1.88 r: Radial coordinate r :Outerradius 1.48 r : Inner radius k: Radii ratio(r/r ) 1.08 i o ∗ h: Lubricant lm fi thickness 0.68 H: Thickness of solid housing −0.10 −0.05 0.00 0.05 0.10 s: Electrical conductivity of the lubricant k=0.35 𝜇 : Viscosity of the lubricant k=0.65 k=0.45 k=0.75 B : Uniform transverse magnetic field applied k=0.55 between the plates 1/2 M: Hartmann Number[B h(s/𝜇) ] Figure 15: Distribution of load bearing capacity with respect to 𝛼 and k. p : Supply Pressure LOAD LOAD LOAD LOAD LOAD Advances in Tribology 7 [12] P.A. Vadher,P.C.Vinodkumar, G. M.Deheri,and R. M. Q: Flow rate Patel, “Behaviour of hydromagnetic squeeze films between two P : Dimensionless supply pressure conducting rough porous circular plates,” Journal of Engineering p: Lubricant pressure Tribology, vol. 222, no. 4, pp. 569–579, 2008. P: Nondimensional pressure [13] J. R. Patel and G. M. Deheri, “Shliomis model based magnetic w: Load carrying capacity u fl id lubrication of a squeeze film in rotating rough curved W: Dimensionless load carrying capacity circular plates, Cerib,” Journal of Science and Technology,vol. 1, h ’: Surface width of lower plate pp.138–150,2013. h ’: Surface width of upper plate [14] R. Patel, G. Deheri, and H. Patel, “Eeff ct of transverse roughness s : Electrical conductivity of lower surface on the performance of a circular step bearing lubricated with a s : Electrical conductivity of upper surface magnetic u fl id,” Annals, vol.8,no.2, pp. 23–30, 2010. 𝜙 (h): Electrical permeability of lower surface(s h /sh) 0 0 0 [15] J.-R. Lin, L.-M. Chu, L.-J. Liang, and P.-H. Lee, “Derivation 𝜙 (h): Electrical permeability of upper surface(s h /sh) 1 1 1 of a modified lubrication equation for hydromagnetic non- 𝜎 : Nondimensional standard deviation Newtonian cylindrical squeeze films and its application to 𝛼 : Dimensionless variance circular plates,” Journal of Engineering Mathematics,vol.77, pp. 𝜀 : Nondimensional skewness. 69–75, 2012. [16] R. M. Patel and G. M. Deheri, “Magnetic u fl id based squeeze film behavior between rotating porous circular plates with Data Availability a concentric circular pocket and surface roughness effects,” International Journal of Applied Mechanics and Engineering,vol. The data used to support the findings of this study are 8, no. 2, pp. 271–277, 2003. available from the corresponding author upon request. [17] P. I. Andharia and G. M. Deheri, “Eeff ct of longitudinal surface roughness on the behaviour of squeeze film in a spherical Conflicts of Interest bearing,” International Journal of Applied Mechanics and Engi- neering,vol.6,no. 4,pp. 885–897, 2001. The authors declare that they have no conflicts of interest. [18] P. I. Andharia and G. Deheri, “Longitudinal roughness effect on magnetic u fl id-based squeeze film between conical plates,” References Industrial Lubrication and Tribology, vol.62,no.5,pp. 285–291, [1] B. C. Majumdar, Introduction to Tribology of Bearing, Wheeler [19] P. I. Andharia and G. M. Deheri, “Performance of magnetic- publishers, Wheeler Co .Ltd, 1985. u fl id-based squeeze film between longitudinally rough elliptical [2] J. Lin, “Static and dynamic characteristics of externally pressur- plates,” ISRN Tribology,vol. 2013, Article ID 482604, 6 pages, ized circular step thrust bearings lubricated with couple stress fluids,” Tribology International, vol.32, no.4,pp.207–216, 1999. [20] M. E. Shimpi and G. M. Deheri, “Combine effect of bearing [3] G. M. Deheri, H.C.Patel, and R.M.Patel, “Performance of deformation and longitudinal roughness on the performance magnetic u fl id based circular step bearings,” Mechanika,vol. 57, of a ferrou fl id based squeeze film together with velocity slip in no.1,pp.22–27,2006. truncated conical plates,” Imperial journal of Interdisciplinary [4] J. L. Neuringer and R. E. Rosensweig, “Magnetic uid fl s,” Physics research, vol.2,no. 8, pp.1423–1430,2016. of Fluids,vol.7,no.12,pp.1927–1937, 1964. [21] J.-R. Lin, “Longitudinal surface roughness effects in magnetic [5] H.Christensen and K.Tonder,“Tribology of rough surfaces: u fl id lubricated journal bearings,” Journal of Marine Science and Parametric study and comparison of lubrication models,” SIN- Technology (Taiwan), vol.24,no.4,pp.711–716,2016. TEF Report,vol.93,no.3, pp. 324–329, 1969. [22] R. M. Patel, G. M. Deheri, and P. A. Vahder, “Hydromagnetic [6] H.Christensen and K.Tonder,“Tribology of rough surfaces: rough porous circular step bearing,” Eastern Academic Journal, Stochastic models of hydrodynamic lubrications,” SINTEF vol. 3, pp. 71–87, 2015. Report, vol.93,no. 3,pp.324–329, 1969. [7] H.Christensen and K.Tonder, “Tribology of rough surfaces: A stochastic model of mixed lubrication,” SINTEF Report,vol. 93, no.3,pp.324–329, 1970. [8] L. L. Ting, “Engagement behavior of lubricated porous annular disks. Part I: squeeze film phase—surface roughness and elastic deformation eeff cts,” Wear, vol.34,no.2, pp. 159–172, 1975. [9] J. Prakash and K. Tiwari, “Roughness eeff cts in porous circular squeeze-plates with arbitrary wall thickness,” Journal of Lubri- cation Technology, vol.105, no.1,pp.90–95,1983. [10] B. L. Prajapati, On Certain eTh oretical Studies in Hydrodynamics And Elastohydrodynamics Lubrication [PhD. thesis],S. P. Uni- versity, V. V. Nagar, India, 1995. [11] P.I. Andharia, J.L.Gupta, and G.M.Deheri, “Eeff ct of transverse surface roughness on the behavior of squeeze film in a spherical bearing,” in Proceedings of the International Confer- ence: Problem of Non-Conventional Bearing Systems (NCBS ’99), 1999. 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A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing

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Copyright © 2018 Jatinkumar V. Adeshara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1687-5923
DOI
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Abstract

Hindawi Advances in Tribology Volume 2018, Article ID 3981087, 7 pages https://doi.org/10.1155/2018/3981087 Research Article A Study of Hydromagnetic Longitudinal Rough Circular Step Bearing 1 2 3 4 Jatinkumar V. Adeshara, M. B. Prajapati, G. M. Deheri , and R. M. Patel Research Scholar, Department of Mathematics, H. N. G. University, Patan, 384 265, Gujarat State, India Head, Department of Mathematics, HNG University, Patan 384 265, Gujarat, India Department of Mathematics, SP University, Vallabh Vidyanagar 388 120, Gujarat, India Department of Mathematics, Gujarat Arts and Science College, Ahmedabad 380 006 Gujarat, India Correspondence should be addressed to R. M. Patel; rmpatel2711@gmail.com Received 13 May 2018; Accepted 10 July 2018; Published 2 September 2018 Academic Editor: Huseyin C¸imenogl ˇ u Copyright © 2018 Jatinkumar V. Adeshara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article discusses the effect of longitudinal roughness on the performance of hydromagnetic squeeze film in circular step bearing. To characterize the random roughness of the bearing surfaces the stochastic model of Christensen and Tonder has been employed. eTh stochastically averaged Reynolds’ type equation is solved using suitable boundary conditions to obtain the pressure distribution and then the load bearing capacity is computed. The results are presented in graphical form. eTh graphical results presented here establish that the hydromagnetic lubrication oeff rs significant help to the longitudinal roughness pattern to enhance the performance of the bearing system. Of course, conductivities of the plates, standard deviation, and the supply pressure contribute towards reducing the negative effect induced by variance (+ve) and skewness (+ve). 1. Introduction of the discussions have considered the stochastic averaging method of Christensen and Tonder [5–7] who presented a It is documented that gears, braking units, hydraulic dampers, stochastic model to evaluate the effect of rough surfaces. skeletal bearings, and synovial joints make use of squeeze (Ting [8], Prakash and Tiwari [9], Prajapati [10], and And- film mechanism. Generally an electrically conducting u fl id haria et al. [11]). with high thermal and electrical conductivity is applied as The hydromagnetic squeeze lm fi s between two conduct- a lubricant for squeeze lm fi to work under such extreme ing rough circular plates are studied by Vadher et al. [12]. circumstances. Also, a use of external magnetic eld fi then It was investigated that the negative effect of roughness got advances the performance of lubrication. minimized up to some extent by applying magnetization. The performance of an oil lubricated circular step bearing Patel and Deheri [13] evaluated the performance of ferrou fl id is analyzed by Majumdar [1]; this study underlines the squeeze lm fi in rotating curved circular plates by using importance of radii ratio. Lin [2] discussed the couple Shliomis model. Here, the outcomes of this article suggested stress effect on the performance of an externally pressurized that Shliomis model based ferrofluid lubrication was rela- circular step bearing. It was established that the couple tively better than the model of Neuringer−Rosensweig [4]. stress uid fl modified the performance of the bearing system. Patel et al. [14] extended the study of Deheri et al. [3] to study Deheri et al. [3] reformed and developed the study of the effect of transverse surface roughness. It was indicated Majumdar [1] to use the magnetization effect by considering that the bearing performance was adversely aeff cted by the a magnetic uid fl as the lubricant, the flow being regulated by Neuringer−Rosensweig’s model [4]. roughness of the bearing surfaces. But the situation was found It is well known that the effect of surface roughness is to be a little enhanced in case of variance (-ve) was con- quite important in different types of bearing systems. Most sidered. A modified lubrication equation for hydromagnetic 2 Advances in Tribology " " 0 0 M ,B 1 1 s, H M ,B 0 0 dr Figure 1: Configuration of bearing system. non-Newtonian cylindrical squeeze lm fi s was presented by (2) The bearing has very low rotational velocity and its Lin et al. [15]. The hydromagnetic non-Newtonian effects effect is neglected for the pressure generation. oer ff ed quite better performance for circular squeeze lm fi s As shown in Figure 1 a thrust load w is applied and the bearing in comparison with the case of a nonconducting Newtonian supports the load without metal to metal contact. The load w lubricant. is supported by the u fl id within the pocket and land. The uid fl Patel and Deheri [16] studied the magnetic u fl id lubrica- escapes in the radial direction through the restrictions by a tion of a squeeze lfi m in transversely rough porous circular land or sill around the recess. plates with concentric circular pocket. Longitudinal rough- In view of the discussions of Christensen and Tonder [5– ness effect has been a matter of discussions in Andharia and 7] the surfaces of bearing are taken to be longitudinally rough. Deheri [17], Andharia and Deheri [18], Andharia and Deheri The thickness h(x) of the lubricant lm fi is given by [19], and Shimpi and Deheri [20]. Recently, Lin [21] examined the ferrou fl id lubrication of a longitudinally rough journal h(x)= h(x)+ h (1) bearing. where h(x) is the mean lm fi thickness and h is the deviation Patel et al. [22] discussed the hydromagnetic lubrication s from the mean lm fi thickness characterizing the random of a rough porous circular step bearing. Hence, it has been roughness surface. h is considered in nature and governed by sought to analyze the effect of longitudinal roughness pattern s the probability density function f(h),− c≤ f(h)≤ c, where on behavior of a hydromagnetic squeeze lm fi in circular step s s c is the maximum deviation from the mean lm fi thickness. bearings. The mean 𝛼 and standard deviation 𝜎 and the measure of symmetry 𝜀 of the random variable h are defined by the 2. Analysis relationships. The configuration of the bearing system is presented in 𝛼= E(h) Figure 1. (2) For the computation of different performance character- 𝜎 = E[(h −𝛼) ] istics, generally the following assumptions are considered: and (1) The recess is deep enough for the pressure in it to be (3) uniform. 𝜀= E[(h −𝛼) ] s Advances in Tribology 3 where E denotes the expected value defined by The concerned Reynolds’ type equation gives the pressure induced ow fl for a circular step bearing as (Majumdar (1985), Patel, Deheri, and Vadher (2015)) E(R)=∫ Rf(h) dh (4) s s −c 2𝜋 r(dp/dr)[(2/ M m(h))[M/2− tanh(M/2)]][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] 0 1 0 1 (5) Q=− 12𝜇 r= r; where p= p; −3 −1 −2 2 2 m(h)= h [1−3𝛼 h +3h (𝜎 +𝛼 ) (7) (6) −3 2 3 − h (𝜀+3𝜎 𝛼+𝛼 )] the governing equation for the film pressure p is given by Using Reynolds’ boundary conditions ln(r/r ) p= p (8) ln(r/r ) i o r= r ; p=0; where in p = ln( ) (9) 𝜋[(2/ M m(h)) M/2− tanh(M/2)][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] r [ ] 0 1 0 1 i Introduction of the dimensionless terms 6 ln(1/k) P = 𝜋((2/ M)/ M∗ h ) M/2− tanh M/2 [(𝜙 +𝜙 +1)/(𝜙 +𝜙 + tanh M/2 / M/2)] ( ( )) [ ( )] ( ( )) ( ) 0 1 0 1 3 ∗ ∗2 ∗2 ∗ ∗2 ∗ ∗3 M∗(h)= m(h) h =(1−3𝛼 +3(𝛼 +𝜎 )− (𝜀 +3𝜎 𝛼 +𝛼 ) 𝛼 =( ) (10) 𝜎 =( ) 𝜀 =( ) k=( ) paves the way for the expression of pressure distribution in 3. Results and Discussions nondimensional form as The representation for dimensionless pressure prole fi is given ln(r/r ) in (11) while the nondimensional load bearing capacity is ∗ o P= P (11) governed by (12). From these equations it is clearly observed ln(r/r ) i o that the load carrying capacity The load bearing capacity w is calculated by integrating the pressure which takes the dimensionless form W∝ P (13) ∗ 2 P ln(1− k ) (12) W= and ln(1/k) 4 Advances in Tribology 5.59 4.34 3.09 1.84 0.59 4.00 6.00 8.00 10.00 12.00 0+1= 0 0+1= 3 0+1= 1 0+1= 4 0+1= 2 Figure 2: Variation of load carrying capacity with respect to M and 𝜙 +𝜙 . 0 1 4.99 3.99 2.99 1.99 0.99 4.00 6.00 8.00 10.00 12.00 ∗=0 ∗=0.15 ∗=0.05 ∗=0.2 ∗=0.1 Figure 3: Distribution of load bearing capacity with respect to M and 𝜎 . p = (14) ((2/M)/m(h))[M/2− tanh(M/2)][(𝜙 +𝜙 +1)/(𝜙 +𝜙 +(tanh(M/2))/(M/2))] 0 1 0 1 This suggests that for a constant ow fl rate the load increases load with respect to𝜙 +𝜙 for different values of standard 0 1 as stochastically averaged squeeze lm fi thickness decreases. deviation, variance, skewness, and radii ratio, respectively. Therefore, the bearing turns out to be self-compensating Here, also the load carrying capacity increases with an provided the flow rate is considered as constant. It is depicted increase in conductivity parameter. Figures 7–10 also tells that from (11) and (12) that the effect of conductivity parameters the load increases sharply up to some extent (𝜙 +𝜙 ≅1.32) 0 1 on the distribution of pressure and load bearing capacity is and then this rate of increase gets slower. determined by It is seen from Figures 11–13 that the standard deviation associated with longitudinal roughness causes increased load 𝜙 +𝜙 + tanh M/2 / M/2 ( ( )) ( ) 0 1 (15) bearing capacity with respect to various parameters such as 𝜙 +𝜙 +1 0 1 variance, skewness, and radii ratio. Besides, variation of load carrying capacity for different values of variance is described which turns to by Figures 14 and 15. From these figures it is noticed that 𝜙 +𝜙 0 1 the load decreases with increase in positive variance while (16) 𝜙 +𝜙 +1 0 1 the variance (− ve) increases it. The trends of load bearing capacity with respect to skewness are similar to that of for large values of M, because tanh(M) ≅ 1and (2/M) ≅ variance which is observed from Figure 16. 0. Further, it is noticed that the pressure and load carrying Form Figures 5 and 9 it is clearly observed that the effect capacity increases with increases in𝜙 +𝜙 because both the 0 1 functions are increasing functions of𝜙 +𝜙 . of𝜀 on the variation of load carrying capacity with respect 0 1 It is clearly depicted from Figures 2–6 that the hydro- to magnetization and conductivity, respectively, remains negligible while this effect of 𝜀 with regards to variance is magnetization parameter induces a sharp increase in the load bearing capacity while Figures 7–10 gives the profile of registered to be nominal as suggested by Figure 14. LOAD LOAD Advances in Tribology 5 5.66 2.06 4.41 1.66 1.26 3.16 0.86 1.91 0+1 0.46 0.66 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=−0.10 ∗=0.05 ∗=−0.05 ∗=0.10 ∗=−0.05 ∗=0.10 ∗=0 ∗=0 Figure 8: Variation of load carrying capacity with respect to 𝜙 +𝜙 Figure 4: Profile of load with respect to M and 𝛼 . 0 1 and𝛼 . 5.00 1.70 4.00 1.45 3.00 1.20 2.00 0.95 0+1 1.00 0.70 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=−0.10 ∗=0.05 ∗=−0.05 ∗=0.10 ∗=−0.05 ∗=0.10 ∗=0 ∗=0 Figure 5: Variation of load carrying capacity with respect to M and Figure 9: Distribution of load bearing capacity with respect to 𝜙 + 𝜀 . 𝜙 and𝜀 . 5.63 2.04 4.38 1.64 3.13 1.24 1.88 0.84 0+1 0.63 0.44 4.00 6.00 8.00 10.00 12.00 0.00 1.00 2.00 3.00 4.00 k=0.35 k=0.65 k=0.35 k=0.65 k=0.45 k=0.75 k=0.45 k=0.75 k=0.55 k=0.55 Figure 6: Distribution of load bearing capacity with respect to M Figure 10: Profile of load with respect to 𝜙 +𝜙 and k. 0 1 and k. 2.06 2.10 1.81 1.75 1.56 1.40 ∗ 1.31 1.05 0+1 1.06 0.70 0.00 0.05 0.10 0.15 0.20 0.00 1.00 2.00 3.00 4.00 ∗=−0.10 ∗=0.05 ∗=0 ∗=0.15 ∗=−0.05 ∗=0.10 ∗=0.05 ∗=0.20 ∗=0 ∗=0.10 Figure 11: Variation of load carrying capacity with respect to 𝜎 and ∗ ∗ Figure 7: Profile of load with respect to 𝜙 +𝜙 and𝜎 . 𝛼 . 0 1 LOAD LOAD LOAD LOAD LOAD LOAD LOAD LOAD 6 Advances in Tribology 2.02 1.91 1.82 1.81 1.62 1.42 1.71 ∗ 1.22 ∗ 1.61 1.02 0.00 0.05 0.10 0.15 0.20 −0.10 −0.05 0.00 0.05 0.10 ∗=−0.10 ∗=0.05 k=0.35 k=0.65 ∗=−0.05 ∗=0.10 k=0.45 k=0.75 ∗=0 k=0.55 ∗ ∗ Figure 12: Distribution of load bearing capacity with respect to 𝜎 Figure 16: Profile of load with respect to 𝜀 and k. and𝜀 . This means that this type of bearing system the suitable 2.27 combination of negatively skewed roughness and negative 2.02 variance maygo a long wayfor better performance of the 1.77 bearing system. 1.52 It is seen that for small as well as large values of M 1.27 the performance of the bearing suffers when the plates are ∗ 1.02 considered to be electrically conducting, in matching with 0.00 0.05 0.10 0.15 0.20 the hydromagnetic case, when the plates are taken to be nonconducting. Further, this can physically be clarified by k=0.35 k=0.65 fringing phenomena which happens when the plates are k=0.45 k=0.75 electrically conducting. Finally, the load is acquired to be k=0.55 more in comparison with the Neuringer Rosensweig model Figure 13: Profile of load with respect to 𝜎 and k. based ferrou fl id squeeze lm. fi 1.87 4. Conclusions 1.67 The trio of hydromagnetization, supply pressure, and stan- dard deviation counters fruitfully the negative effect induced 1.47 by positively skewed roughness and variance (+ve) in the 1.27 case of negatively skewed roughness when variance comes ∗ out to be negative. Of course, in bettering the bearing 1.07 performancethe plateconductivities and radiiratio may play −0.10 −0.05 0.00 0.05 0.10 an important role. Needless to say, the longitudinal roughness ∗=−0.10 ∗=0.05 parameter behaves in a little better way as compared to ∗=−0.05 ∗=0.10 the transverse roughness. Therefore, it becomes very much ∗=0 essential that the longitudinal roughness aspect must be Figure 14: Variation of load carrying capacity with respect to 𝛼 and given due consideration from longevity point of view while 𝜀 . designing this type of bearing system. 2.28 Nomenclature 1.88 r: Radial coordinate r :Outerradius 1.48 r : Inner radius k: Radii ratio(r/r ) 1.08 i o ∗ h: Lubricant lm fi thickness 0.68 H: Thickness of solid housing −0.10 −0.05 0.00 0.05 0.10 s: Electrical conductivity of the lubricant k=0.35 𝜇 : Viscosity of the lubricant k=0.65 k=0.45 k=0.75 B : Uniform transverse magnetic field applied k=0.55 between the plates 1/2 M: Hartmann Number[B h(s/𝜇) ] Figure 15: Distribution of load bearing capacity with respect to 𝛼 and k. p : Supply Pressure LOAD LOAD LOAD LOAD LOAD Advances in Tribology 7 [12] P.A. Vadher,P.C.Vinodkumar, G. M.Deheri,and R. M. Q: Flow rate Patel, “Behaviour of hydromagnetic squeeze films between two P : Dimensionless supply pressure conducting rough porous circular plates,” Journal of Engineering p: Lubricant pressure Tribology, vol. 222, no. 4, pp. 569–579, 2008. P: Nondimensional pressure [13] J. R. Patel and G. M. Deheri, “Shliomis model based magnetic w: Load carrying capacity u fl id lubrication of a squeeze film in rotating rough curved W: Dimensionless load carrying capacity circular plates, Cerib,” Journal of Science and Technology,vol. 1, h ’: Surface width of lower plate pp.138–150,2013. h ’: Surface width of upper plate [14] R. Patel, G. Deheri, and H. Patel, “Eeff ct of transverse roughness s : Electrical conductivity of lower surface on the performance of a circular step bearing lubricated with a s : Electrical conductivity of upper surface magnetic u fl id,” Annals, vol.8,no.2, pp. 23–30, 2010. 𝜙 (h): Electrical permeability of lower surface(s h /sh) 0 0 0 [15] J.-R. Lin, L.-M. Chu, L.-J. Liang, and P.-H. Lee, “Derivation 𝜙 (h): Electrical permeability of upper surface(s h /sh) 1 1 1 of a modified lubrication equation for hydromagnetic non- 𝜎 : Nondimensional standard deviation Newtonian cylindrical squeeze films and its application to 𝛼 : Dimensionless variance circular plates,” Journal of Engineering Mathematics,vol.77, pp. 𝜀 : Nondimensional skewness. 69–75, 2012. [16] R. M. Patel and G. M. Deheri, “Magnetic u fl id based squeeze film behavior between rotating porous circular plates with Data Availability a concentric circular pocket and surface roughness effects,” International Journal of Applied Mechanics and Engineering,vol. The data used to support the findings of this study are 8, no. 2, pp. 271–277, 2003. available from the corresponding author upon request. [17] P. I. Andharia and G. M. Deheri, “Eeff ct of longitudinal surface roughness on the behaviour of squeeze film in a spherical Conflicts of Interest bearing,” International Journal of Applied Mechanics and Engi- neering,vol.6,no. 4,pp. 885–897, 2001. The authors declare that they have no conflicts of interest. [18] P. I. Andharia and G. Deheri, “Longitudinal roughness effect on magnetic u fl id-based squeeze film between conical plates,” References Industrial Lubrication and Tribology, vol.62,no.5,pp. 285–291, [1] B. C. Majumdar, Introduction to Tribology of Bearing, Wheeler [19] P. I. Andharia and G. M. Deheri, “Performance of magnetic- publishers, Wheeler Co .Ltd, 1985. u fl id-based squeeze film between longitudinally rough elliptical [2] J. Lin, “Static and dynamic characteristics of externally pressur- plates,” ISRN Tribology,vol. 2013, Article ID 482604, 6 pages, ized circular step thrust bearings lubricated with couple stress fluids,” Tribology International, vol.32, no.4,pp.207–216, 1999. [20] M. E. Shimpi and G. M. Deheri, “Combine effect of bearing [3] G. M. Deheri, H.C.Patel, and R.M.Patel, “Performance of deformation and longitudinal roughness on the performance magnetic u fl id based circular step bearings,” Mechanika,vol. 57, of a ferrou fl id based squeeze film together with velocity slip in no.1,pp.22–27,2006. truncated conical plates,” Imperial journal of Interdisciplinary [4] J. L. Neuringer and R. E. Rosensweig, “Magnetic uid fl s,” Physics research, vol.2,no. 8, pp.1423–1430,2016. of Fluids,vol.7,no.12,pp.1927–1937, 1964. [21] J.-R. Lin, “Longitudinal surface roughness effects in magnetic [5] H.Christensen and K.Tonder,“Tribology of rough surfaces: u fl id lubricated journal bearings,” Journal of Marine Science and Parametric study and comparison of lubrication models,” SIN- Technology (Taiwan), vol.24,no.4,pp.711–716,2016. TEF Report,vol.93,no.3, pp. 324–329, 1969. [22] R. M. Patel, G. M. Deheri, and P. A. Vahder, “Hydromagnetic [6] H.Christensen and K.Tonder,“Tribology of rough surfaces: rough porous circular step bearing,” Eastern Academic Journal, Stochastic models of hydrodynamic lubrications,” SINTEF vol. 3, pp. 71–87, 2015. Report, vol.93,no. 3,pp.324–329, 1969. [7] H.Christensen and K.Tonder, “Tribology of rough surfaces: A stochastic model of mixed lubrication,” SINTEF Report,vol. 93, no.3,pp.324–329, 1970. [8] L. L. Ting, “Engagement behavior of lubricated porous annular disks. Part I: squeeze film phase—surface roughness and elastic deformation eeff cts,” Wear, vol.34,no.2, pp. 159–172, 1975. [9] J. Prakash and K. Tiwari, “Roughness eeff cts in porous circular squeeze-plates with arbitrary wall thickness,” Journal of Lubri- cation Technology, vol.105, no.1,pp.90–95,1983. [10] B. L. Prajapati, On Certain eTh oretical Studies in Hydrodynamics And Elastohydrodynamics Lubrication [PhD. thesis],S. P. Uni- versity, V. V. Nagar, India, 1995. [11] P.I. Andharia, J.L.Gupta, and G.M.Deheri, “Eeff ct of transverse surface roughness on the behavior of squeeze film in a spherical bearing,” in Proceedings of the International Confer- ence: Problem of Non-Conventional Bearing Systems (NCBS ’99), 1999. 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