Acta Polytechnica DOI:10.14311/AP.2020.60.0259 Acta Polytechnica 60(3):259–267, 2020 © Czech Technical University in Prague, 2020 available online at https://ojs.cvut.cz/ojs/index.php/ap NEURINGER-ROSEINWEIG MODEL BASED LONGITUDINALLY ROUGH POROUS CIRCULAR STEPPED PLATES IN THE EXISTENCE OF COUPLE STRESS Yogini Devendrasinh Vashia, ∗, Rakesh Manilal Patelb, Gunamani Biswanath Deheric a Gujarat Technological University, Alpha College of Engineering and Technology, Kalol, 382721 Gujarat, India b Gujrat University, Gujarat Arts and Science College, Ahmedabad, 380006 Gujarat, India c Sardar Patel University, Department of Mathematics Vallabh vidyanagar, 388120 Gujarat, India ∗ corresponding author: yogini.vashi@gmail.com Abstract. This study intents to scrutinize the impact of ferrofluid in the presence of couple stress for longitudinally rough porous circular stepped plates. The influence of longitudinal surface roughness is developed using the stochastic model of Christensen and Tonder for nonzero mean, variance and skewness. Neuringer-Roseinweig model is adopted for the influence of ferrofluid. The couple stress effect is characterized by Stoke’s micro continuum theory. The modified Reynolds’ type equation is stochastically averaged and solved by no-slip boundary conditions. The closed form solutions for load bearing capacity and film pressure are obtained as a function of different parameters and plotted graphically. It is perceived that the load capacity gets elevated owing to the combined influence of magnetization and couple stress when the proper choice of roughness parameters (negatively skewed, standard deviation) is in place. Porosity and roughness (positively skewed) adversely affect bearing’s performance. The graphical and tabular analysis shows that there is a significant growth in load bearing capacity compared to the conventional lubricant case. Keywords: Load bearing capacity, ferrofluid, longitudinal roughness, circular stepped plates. 1. Introduction A ferrofluid is a liquid that contains a colloidal sus- pension of ferromagnetic particles and that becomes strongly magnetized in the existence of an external magnetic field. The use of ferrofluid as a lubricant can be found in bearings, loudspeakers, dampers, sensor as well as in biomedical instruments. Several inves- tigators have also attempted to discover its use as a lubricant in squeeze film bearing structures. Bhat and Deheri [1] studied squeeze film characteristic in the curved circular disk by considering the magnetic effect and found that bearing’s load capacity is en- hanced due to the magnetization parameter. Shah [2] analysed the impact of ferrofluid lubrication in step bearing. Kumar et. al. [3] considered the influence of ferrofluid on spherical and conical bearings using perturbation analysis. Shah and Bhat studied [4] the impact of magnetic fluid on curved annular plates. They considered the revolution of magnetic particles and its magnetic moments. Their study revealed that the load capacity and response time of the bearing improved due to Langevin’s parameter. Agrawal [5] examined the impact of porous inclined slider bearing with magnetic fluid and deduced that bearing’s life span is longer compared to the viscous porous inclined slider bearing. Additives have been added in the fluid to create the flow properties and to enhance the lubricating quali- ties. Couple stress and micropolar fluids are examples of such types of fluids, which have become more im- portant for current industrial materials. Extensive studies have been carried out to describe the impor- tance of the couple stress fluid in different bearing geometries. Ramanaiah and Sarkar [6] analysed the upshot of couple stress for the thrust bearing. Lin [7] made a theoretical study of a squeeze film based finite journal bearing with a couple stress fluid and con- cluded that squeeze film-based finite journal bearing’s characteristics is enhanced due to rheological proper- ties of the couple stress fluid. Maiti [8] analysed the performance characteristic of the composite and step slider bearing with micropolar fluid. He has derived the expression of the load supporting capacity and skin friction using numerical computation and found that the micropolar fluid offers better load support- ing capacity compared to a Newtonian fluid. Lin et al. [9] studied the combined performance of couple stress and convective inertia on wide parallel plates and concluded that there is an improvement in the load-carrying capacity and in the response time due to a combined effect of couple stress and convective inertia. Naduvinamani and Siddangouda [10] con- sidered a couple stress fluid to study the impact of the squeeze film lubrication in circular stepped plates. In this investigation, the couple stress impact is gov- erned by Stokes microcontinuum [11] theory and this study discovered the advantages of couple stress fluid compared to Newtonian lubricants, such as improved 259 https://doi.org/10.14311/AP.2020.60.0259 https://ojs.cvut.cz/ojs/index.php/ap Y. D. Vashi, R. M. Patel, G. B. Deheri Acta Polytechnica bearing’s load capacity, reduced coefficient of friction and growth in squeeze film time. However, it is well known that after having some run-in wear, bearing surfaces develop some roughness so, surface roughness and its impact on bearing perfor- mance have been deliberated by various investigators. Some mathematical models have been anticipated in the derivation of Reynolds type equations, which accounts for the surface roughness effect. Among all these models, the stochastic approach given by Christensen and Tonder [12–14] is used very widely. Many investigators have studied the combined influ- ence of surface roughness and ferrofluid for a distinct porous bearing configuration [15–20]. All these scruti- nies discovered that the load capacity of the bearing is improved due to ferrofluid and negatively skewed roughness. Andharia and Deheri [21] studied the ef- fect of ferrofluid for longitudinally rough elliptical plates. Patel and Deheri [22] analysed the combined influence of ferrofluid and slip velocity for longitu- dinally rough conical plates and found that load ca- pacity gets enhanced due to the magnetization and standard deviation. Shah and Patel [23] studied the influence of ferrofluid for a rotating sphere with a radially rough lower flat plate and shown that the overall performance of the sphere plate is improved due to a variable magnetic field. Lin [24] studied the influence of a magnetic fluid on journal bearings with longitudinal surface roughness and shown that the effects of longitudinal roughness suggest a growth in the load capacity along with a reduction in the fric- tion parameter and the attitude angle compared to non-magnetic smooth surface case. The joint impact of surface roughness and couple stress on different bearing geometry is investigated by [25–27] and their investigations confirmed that the load capacity and squeeze film time increase due to the non-Newtonian behaviour of fluid compared to Newtonian case. So, it is supposed to examine the influence of fer- rofluid and longitudinal surface roughness on porous circular stepped plates in the presence of couple stress. 2. Analysis Figure 1 displays the bearing geometry. The lower plate has porous facing, which remains fixed while the upper plate is moving with a squeeze velocity V . The fluid region is filled with incompressible ferrofluid with a couple stress effect. To characterize the surface roughness, the film thickness is taken as Hi = hi + hs for i = 1, 2 where, hi represents the mean film thickness and hs is the quantity due to surface roughness calculated from mean level and is considered as a randomly chang- ing quantity of non-zero mean. Additionaly, hs is considered to be stochastic in nature having a polyno- mial form of a probability distribution function given by Christensen and Tonder [12–14]. The roughness parameters are stated by the relations α = E(hs) σ2 = E [ (hs −α) 2 ] ε = E [ (hs −α) 3 ] Where the expectancy operator E(•) is described as E(•) = ∫ ∞ −∞ (•)f(hs)dhs Through the traditional assumptions of hydrody- namic lubrication, the governing Reynolds type equa- tion for the film pressure turns out be [10] dpi dr = −6µV r Si(Hi, l) (1) Si(hi, l) = H3i + 12φH − 12l 2Hi + 24l3 tan h ( Hi 2l ) (2) for smooth bearing. To derive the longitudinal surface roughness, we adopted the stochastic approach given by Christensen and Tonder [12–14] in equation (1). dpi dr = −6µV r gi(hi,α,σ,ε, l) (3) where, gi(hi,α,σ,ε, l) = 1 E ( H−3i ) + 12φH− − 12l2 1 E ( H−1i ) + 24l3 tan h   1E(H−1i ) 2l   (4) where, gi(hi,α,σ,ε, l) = 1 mi(hi,α,σ,ε) + 12φH− −12l2 1 ni(hi,α,σ,ε) + 24l3 tan h   1 ni(hi,α,σ,ε) 2l   (5) mi(hi,α,σ,ε) = h−3i ( 1 − 3αh−1i + 6h −2 i ( σ2 + α2 ) −10h−3i ( 3σ2α + α3 + ε )) (6) ni(hi,α,σ,ε) = h−1i ( 1 −αh−1i + h −2 i ( σ2 + α2 ) −h−3i ( 3σ2α + α3 + ε )) (7) Neuringer-Rosenswein [28] intended a simple model to define the stable flow of ferrofluids in the existence of 260 vol. 60 no. 3/2020 Neuringer-Roseinweig model based longitudinally. . . Figure 1. Physical configuration of circular step plates. gradually varying magnetic fields. The model involves the following equations ρ(q ·∇)q = −∇p + µ∇2q + µ0(M ·∇)H (8) ∇·q = 0 (9) ∇×H = 0 (10) M = µH (11) ∇· (H + M) = 0 (12) By using equation (10) and (11) in equation (8), it reduces to ρ(q ·∇)q = −∇ ( p− µ0µ 2 H2 ) + µ∇2q (13) Equation (13) represents that additional pressure term 1 2µ0µH 2 is present in the equation of motion when ferrofluid is used as a lubricant. Therefore, in the context of Neuringer - Rosensweig [28] model, equation (3) transfer to d dr [ pi − 0.5µ0µH2 ] = −6µV r gi(hi,α,σ,ε, l) (14) in equation (14), H2 denotes the magnetic field’s magnitude and is given by H2 = A(R−r)(r −KR) (15) where A is an appropriate constant reliant on the material to yield a field of expected magnetic field strength. Where, • hi = h1 minimum film thickness in the film region 0 ≤ r ≤ KR and • hi = h2 maximum film thickness in the film region KR ≤ r ≤ R. The related fluid film pressure boundary conditions are p1 = p2 at r = KR and p2 = 0 at r = R (16) The integration of equation (14) with respect to r with the boundary conditions given in expression (16) yields the pressure in both the film regions p1 = 3µV g1 (K2R2 −r2) + 3µV g2 (R2 −R2K2)+ + 0.5µ0µH2 (17) p2 = 3µV g2 (R2 −r2) + 0.5µ0µH2 (18) where, g1 = 1 m1(h1,α,σ,ε) + 12φH −12l2 1 n1(h1,α,σ,ε) + + 24l3 tan h   1 n1(h1,α,σ,ε) 2l   (19) 261 Y. D. Vashi, R. M. Patel, G. B. Deheri Acta Polytechnica g2 = 1 m2(h2,α,σ,ε) + 12φH −12l2 1 n2(h2,α,σ,ε) + + 24l3 tan h   1 n2(h2,α,σ,ε) 2l   (20) Integrating the expression (17) and ( 18), the load bearing capacity w is obtained as w = 2π ∫ KR 0 p1rdr + 2π ∫ R KR p2rdr (21) which yields w = πR40.5Aµ0µ 6 (1 − 2K)+ + 3πµV R4 2 [ K4 g1 + (1 −K4) g2 ] (22) By making the use of following dimensionless variables in equation (22). µ∗ = − µ0µAh 3 2 µV ,H∗ = h1 h2 , l∗ = 2l h2 ,α∗ α h2 , σ∗ = σ h2 ,ε∗ = ε h32 ,ψ = φH h32 (23) Expression (22) transfers to dimensionless load capac- ity as follows. w = 2wh32 3πµV R4 = µ∗(2K − 1) 18 + [ K4 G1 + (1 −K4) G2 ] (24) where, G1 = h32 g1 = 1 M1(H∗,α∗,σ∗,ε∗) + 12ψ− − 3l∗2 N1(H∗,α∗,σ∗,ε∗) + + 3l∗3 tan h ( 1 N1(H∗,α∗,σ∗,ε∗)l∗ ) (25) G2 = h32 g2 = 1 M2(α∗,σ∗,ε∗) + 12ψ− − 3l∗2 N2(α∗,σ∗,ε∗) + + 3l∗3 tan h ( 1 N2(α∗,σ∗,ε∗)l∗ ) (26) where, M1(H∗,α∗,σ∗,ε∗) = H∗−3(1 − 3α∗H∗−1+ + 6H∗−2(σ∗2 + α∗2)− − 10H∗−3(3σ∗2α∗ + α∗3 + ε∗)) (27) N1(H∗,α∗,σ∗,ε∗) = H∗−1(1 −α∗H∗−1+ + H∗−2(σ∗2 + α∗2)− −H∗−3(3σ∗2α∗ + α∗3 + ε∗)) (28) 0.40 0.64 0.88 1.12 1.36 1.60 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* K=0.55 K=0.65 K=0.75 K=0.85 K=0.95 Figure 2. Profile of w for the combination of µ∗ with various values K, with H∗ = 2.10, σ∗ = 0.05, α∗ = −0.025, ε∗ = −0.025, ψ = 0.001 and l∗ = 0.3. 1.43 1.51 1.58 1.66 1.73 1.81 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* s*= 0 s*= 0.05 s*= 0.10 s*= 0.15 s*= 0.2 Figure 3. Profile of w for the combination of µ∗ with various values σ∗, with H∗ = 2.10, K = 0.65, α∗ = −0.025, ε∗ = −0.025, ψ = 0.001 and l∗ = 0.3. M2(α∗,σ∗,ε∗) = (1 − 3α∗ + 6(σ∗2 + α∗2)− − 10(3σ∗2α∗ + α∗3 + ε∗)) (29) N2(α∗,σ∗,ε∗) = (1 −α∗ + (σ∗2 + α∗2)− − (3σ∗2α∗ + α∗3 + ε∗)) (30) 3. Results and discussion This study examined the performance characteristic of longitudinally rough porous circular stepped plates with ferrofluid in the presence of couple stress effect. The equation (17), equation (18) and equation (24) represent the closed form solution for pressure and load capacity. Equation (24) suggests that load ca- pacity is enhanced by µ ∗(2K−1) 18 times as compared to a conventional lubricant based bearing system. Also, the comparison is made between the current results with the results obtained by [10], The effect of longitu- dinal surface roughness is described by the parameter σ∗,α∗,ε∗. The variation of w with regards to µ∗ for distinct values of K,σ∗,α∗,ε∗,ψ,l∗ is displayed in Figures 2-7. One can notice from all these Figures that the effect of µ∗ is an improvement of the load bearing capacity. From the physical perspective, µ∗ increases the viscosity of the lubricant, which results in improved pressure and load capacity. Fig. 7 shows that the influence of µ∗ is more emphasized for a larger value of l∗. 262 vol. 60 no. 3/2020 Neuringer-Roseinweig model based longitudinally. . . 1.19 1.27 1.34 1.42 1.49 1.57 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* a*=-0.05 a*=-0.025 a*=0 a*=0.025 a*= 0.05 Figure 4. Profile of w for the combination of µ∗ with various values α∗, with H∗ = 2.10, K = 0.65, σ∗ = 0.05, ε∗ = −0.025, ψ = 0.001 and l∗ = 0.3. 0.55 0.80 1.06 1.31 1.57 1.82 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* e*=-0.05 e*=-0.025 e*=0 e*=0.025 e*= 0.05 Figure 5. Profile of w for the combination of µ∗ with various values ε∗, with H∗ = 2.10, K = 0.65, σ∗ = 0.05, α∗ = −0.025, ψ = 0.001 and l∗ = 0.3. 0.47 0.67 0.88 1.08 1.29 1.49 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* y=0 y=0.0001 y=0.001 y=0.01 y=0.1 Figure 6. Change in w plotted for µ∗ with various val- ues of ψ, with H∗ = 2.10, K = 0.65, σ∗ = 0.05, α∗ = −0.025, ε∗ = −0.025 and l∗ = 0.3. 1.13 1.34 1.54 1.75 1.95 2.16 0.00 0.05 0.10 0.15 0.20 0.25 L O A D m* l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5 Figure 7. Variation of w plotted for µ∗ with vari- ous values of l∗, with H∗ = 2.10, K = 0.65, σ∗ = 0.05, α∗ = −0.025, ε∗ = −0.025 and ψ = 0.001. 0.52 0.93 1.35 1.76 2.18 2.59 0.55 0.63 0.71 0.79 0.87 0.95 L O A D K s*= 0 s*= 0.05 s*= 0.10 s*= 0.15 s*= 0.2 Figure 8. Profile of w referenced to K for vari- ous values of σ∗, with H∗ = 2.10, µ∗ = 0.15, α∗ = −0.025, ε∗ = −0.025, ψ = 0.001 and l∗ = 0.3. 0.36 0.63 0.90 1.18 1.45 1.72 0.55 0.63 0.71 0.79 0.87 0.95 L O A D K a*=-0.05 a*=-0.025 a*=0 a*=0.025 a*= 0.05 Figure 9. Trends of w referenced to K for various val- ues of α∗, with H∗ = 2.10, µ∗ = 0.15, σ∗ = 0.05, ε∗ = −0.025, ψ = 0.001 and l∗ = 0.3. Figures 8 - 12 represent the influence of K on w for various values of σ∗,α∗,ε∗,ψ and l∗. From all these Figures, one can conclude that w decreases as the value of K increases. Also, Figure 12 suggests that the decrease of w is insignificant for higher values of l∗. Distribution of w with regard to H∗ for distinct values of σ∗,α∗,ε∗,ψ and l∗ is presented in Figures 13 - 17. It can be seen that w decreases as the value of H∗ increases and that the decrease of w is minimal when H∗ surpasses the value of 2.26. Figures 18 - 21 characterize the positive impact of σ∗ on the load capacity for various values of α∗,ε∗,ψ,l∗. All these Figures show that there is a sharp growth in w as the values of σ∗ increase. In the case of longitudinal surface roughness, the parameter σ∗ plays a crucial role as it supports the motion of the lubricant, which results in an increased pressure, and therefore load capacity. Figure 21 suggests that the maximum load is registered when l∗ = 0.5 and σ = 0.2. The positive influence of ε∗ on w with regards to ψ and l∗ can be seen from Figures 22 - 23. From both these Figures, one can observe that a negative increase in the values of ε∗ rises the dimensionless load capacity of bearing. So, the optimistic impact of ε∗ may be accordingly considered while designing the bearing structure. Figures 6, 11, 16, 20 and 22 clearly show that the initial impact of porous facing is insignificant up to a value of ψ = 0.001. 263 Y. D. Vashi, R. M. Patel, G. B. Deheri Acta Polytechnica K l∗ w for the present study with  µ∗ = 0.15, σ∗ = 0.15,α∗ = −0.025, ε = −0.025, ψ = 0.001, H∗ = 2.10   w obtained from [10] Relative percentage growth in w (Rw) 0.55 0.1 1.2578614535 0.9436490018 33.30 0.3 1.5970615126 1.1309267877 41.22 0.5 2.3701103455 1.4970075375 58.32 0.75 0.1 0.9786901999 0.7369511955 32.80 0.3 1.2353809760 0.8791740326 40.52 0.5 1.8198623988 1.1571299684 57.27 0.95 0.1 0.3563430762 0.2791647245 27.65 0.3 0.4302954180 0.3216015680 33.80 0.5 0.5971448849 0.4043820041 47.67 Table 1. Change in w and Rw for distinct values of K and l∗. H l∗ w for the present study with  µ∗ = 0.15, σ∗ = 0.15,α∗ = −0.025, ε = −0.025, ψ = 0.001, K = 0.65   w obtained from [10] Relative percentage growth in w (Rw) 1.3 0.1 1.23184347 0.92689311 32.90 0.3 1.55161185 1.10583213 40.31 0.5 2.27664670 1.45544961 56.42 2.1 0.1 1.15524444 0.86369031 33.76 0.3 1.46252673 1.03353911 41.51 0.5 2.16263041 1.36552978 58.37 2.9 0.1 1.14218948 0.85163399 34.12 0.3 1.44862282 1.02073256 41.92 0.5 2.14709608 1.35128724 58.89 Table 2. Change in w and Rw for distinct values of H∗ and l∗. 0.21 0.57 0.93 1.30 1.66 2.02 0.55 0.63 0.71 0.79 0.87 0.95 L O A D K e*=-0.05 e*=-0.025 e*=0 e*=0.025 e*= 0.05 Figure 10. Trends of w referenced to K for vari- ous values of ε∗, with H∗ = 2.10, µ∗ = 0.15, σ∗ = 0.05, α∗ = −0.025, ψ = 0.001 and l∗ = 0.3. Tables 1 - 2 clearly show that the bearing’s per- formance improves due to the combined influence of ferrofluid and couple stress when compared to the results obtained from [10]. It can be seen from Ta- ble 1 that there is a growth of about 58 % in w when K = 0.55 and l∗ = 0.5. Also, from Table 2, it can be seen that a relative increase in w is around 59 % when H∗ = 2.9 and l∗ = 0.5. 0.19 0.48 0.77 1.06 1.35 1.64 0.55 0.63 0.71 0.79 0.87 0.95 L O A D K y=0 y=0.0001 y=0.001 y=0.01 y=0.1 Figure 11. Trends of w referenced to K for vari- ous values of ψ∗, with H∗ = 2.10, µ∗ = 0.15, σ∗ = 0.05, ε∗ = −0.025, α∗ = −0.025 and l∗ = 0.3. 4. Conclusion The combined impact of surface roughness and fer- rofluid for porous circular stepped plates in the pres- ence of couple stress is analysed. The graphical and tabular results show that the combined impact of fer- rofluid and couple stress improves the load bearing capacity considerably. Furthermore, this improvement in load bearing capacity is almost 58 % higher when 264 vol. 60 no. 3/2020 Neuringer-Roseinweig model based longitudinally. . . 0.34 0.75 1.16 1.56 1.97 2.38 0.55 0.63 0.71 0.79 0.87 0.95 L O A D K l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5 Figure 12. Trends of w referenced to K for vari- ous values of l∗, with H∗ = 2.10, µ∗ = 0.15, σ∗ = 0.05, ε∗ = −0.025, α∗ = −0.025 and ψ = 0.001. 1.41 1.51 1.61 1.70 1.80 1.90 1.30 1.62 1.94 2.26 2.58 2.90 L O A D H* s*= 0 s*= 0.05 s*= 0.10 s*= 0.15 s*= 0.2 Figure 13. Trends of w referenced to H∗ for vari- ous values of σ∗, with K = 0.65, ψ = 0.001, µ∗ = 0.15, ε∗ = −0.025, α∗ = −0.025 and l∗ = 0.3. 1.17 1.27 1.37 1.46 1.56 1.66 1.30 1.62 1.94 2.26 2.58 2.90 L O A D H* a*=-0.05 a*=-0.025 a*=0 a*=0.025 a*= 0.05 Figure 14. Trends of w referenced to H∗ for vari- ous values of α∗, with K = 0.65, ψ = 0.001, µ∗ = 0.15, ε∗ = −0.025, σ∗ = 0.05 and l∗ = 0.3. 0.54 0.82 1.09 1.37 1.64 1.92 1.30 1.62 1.94 2.26 2.58 2.90 L O A D H* e*=-0.05 e*=-0.025 e*=0 e*=0.025 e*= 0.05 Figure 15. Trends of w referenced to H∗ for var- ious values of ε∗, with K = 0.65, ψ = 0.001, µ∗ = 0.15, α∗ = −0.025, σ∗ = 0.05 and l∗ = 0.3. 0.46 0.68 0.91 1.13 1.36 1.58 1.30 1.62 1.94 2.26 2.58 2.90 L O A D H* y=0 y=0.0001 y=0.001 y=0.01 y=0.1 Figure 16. Trends of w referenced to H∗ for various values of ψ, with K = 0.65, α∗ = −0.025, µ∗ = 0.15, ε∗ = −0.025, σ∗ = 0.05 and l∗ = 0.3. 1.12 1.35 1.58 1.82 2.05 2.28 1.30 1.62 1.94 2.26 2.58 2.90 L O A D H* l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5 Figure 17. Trends of w referenced to H∗ for various values of l∗, with K = 0.65, α∗ = −0.025, µ∗ = 0.15, ε∗ = −0.025, σ∗ = 0.05 and ψ = 0.001. 1.18 1.34 1.49 1.65 1.80 1.96 0.00 0.05 0.10 0.15 0.20 L O A D s* a*=-0.05 a*=-0.025 a*=0 a*=0.025 a*= 0.05 Figure 18. Distribution of w referenced to σ∗ for various values of α∗, with K = 0.65, ψ = 0.001, µ∗ = 0.15, ε∗ = −0.025, H∗ = 2.10 and l∗ = 0.3. 0.55 0.88 1.21 1.54 1.87 2.20 0.00 0.05 0.10 0.15 0.20 L O A D s* e*=-0.05 e*=-0.025 e*=0 e*=0.025 e*= 0.5 Figure 19. Distribution of w referenced to σ∗ for various values of ε∗, with K = 0.65, ψ = 0.001, µ∗ = 0.15, α∗ = −0.025, H∗ = 2.10 and l∗ = 0.3. 265 Y. D. Vashi, R. M. Patel, G. B. Deheri Acta Polytechnica 0.47 0.75 1.02 1.30 1.57 1.85 0.00 0.05 0.10 0.15 0.20 L O A D s* y=0 y=0.0001 y=0.001 y=0.01 y=0.1 Figure 20. Distribution of w referenced to σ∗ for various values of ψ, with K = 0.65, α∗ = −0.025, µ∗ = 0.15, ε∗ = −0.025, H∗ = 2.10 and l∗ = 0.3. 1.12 1.47 1.82 2.17 2.52 2.87 0.00 0.05 0.10 0.15 0.20 L O A D s* l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5 Figure 21. Distribution of w referenced to σ∗ for various values of l∗, with K = 0.65, α∗ = −0.025, µ∗ = 0.15, ε∗ = −0.025, H∗ = 2.10 and ψ = 0.001. 0.32 0.63 0.94 1.26 1.57 1.88 -0.05 -0.03 -0.01 0.01 0.03 0.05 L O A D e* y=0 y=0.0001 y=0.001 y=0.01 y=0.1 Figure 22. Distribution of w referenced to ε∗ for various values of ψ, with K = 0.65, α∗ = −0.025, µ∗ = 0.05, σ∗ = 0.15, H∗ = 2.10 and l∗ = 0.3. 0.50 1.01 1.53 2.04 2.56 3.07 -0.05 -0.03 -0.01 0.01 0.03 0.05 L O A D e* l*=0.1 l*=0.2 l*=0.3 l*=0.4 l*=0.5 Figure 23. Distribution of w referenced to ε∗ for various values of l∗, with K = 0.65, α∗ = −0.025, µ∗ = 0.15, σ∗ = 0.05, H∗ = 2.10 and ψ = 0.001. compared to the results obtained from [10]. Also, the development in load capacity is enhanced in the pres- ence of a standard deviation and negatively skewed roughness with a couple stress parameter. The reduc- tion in load is due to the porous facing and positively skewed roughness can be compensated by employing the ferrofluid as a lubricant with the suitable choice of a couple stress parameter through which the life span of a circular step bearing can be boosted. Acknowledgements The authors acknowledge the valuable remarks and sug- gestions of the reviewers, which have contributed to the development of the presentation and organization of the manuscript. List of symbols H porous facing thickness [mm] H∗ dimensionless fluid film thickness H magnetic field vector h1 maximum fluid film thickness [mm] h2 minimum fluid film thickness [mm] KR step location (0 < K < 1) l couple stress parameter (√ η µ ) l∗ dimensionless couple stress parameter M magnetization vector p1 film pressure for the region (0 < r < KR) p2 film pressure for the region (KR < r < R) q fluid velocity vector r radial coordinate R radius of the circular plate V squeeze velocity w load capacity of bearing [N] w dimensionless load capacity α variance [mm] α∗ dimensionless variance ε skewness [mm3] ε∗ dimensionless skewness η couple stress constant of the lubricant µ fluid viscosity [N S/m2] µ0 permeability of the free space µ magnetic susceptibility of particle µ∗ dimensionless magnetization parameter σ standard deviation [mm] σ∗ dimensionless standard deviation φ porous facing permeability ψ porosity of porous facing References [1] M. Bhat, G. Deheri. 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Physics of fluids 7(12):1927 – 1937, 1964. 267 http://dx.doi.org/10.1016/0020-7225(92)90008-5 http://dx.doi.org/10.1007/s10665-004-1770-9 http://dx.doi.org/10.1016/0043-1648(86)90023-2 http://dx.doi.org/10.1016/0043-1648(78)90229-6 http://dx.doi.org/10.1016/S0301-679X(98)00022-X http://dx.doi.org/10.1143/jjap.12.1058 http://dx.doi.org/10.1590/S1678-58782009000100004 http://dx.doi.org/10.1007/978-3-642-82351-0_4 http://dx.doi.org/10.15282/jmes.4.2013.17.0050 http://dx.doi.org/10.29252/jafm.11.03.27854 http://dx.doi.org/10.14311/AP.2019.59.0144 http://dx.doi.org/10.1080/17515831.2017.1418280 http://dx.doi.org/10.5402/2013/482604 http://dx.doi.org/10.5937/jsscm1602018P http://dx.doi.org/10.1007/s11012-015-0337-3 http://dx.doi.org/10.6119/JMST-016-0201-1 http://dx.doi.org/10.1016/j.triboint.2006.07.003 http://dx.doi.org/10.1002/ls.24 Acta Polytechnica 60(3):259–267, 2020 1 Introduction 2 Analysis 3 Results and discussion 4 Conclusion Acknowledgements List of symbols References