Magnetic Fluid Based Squeeze Film behavior between curved circular Plates and Surface Roughness Effect

Transcription

1 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 Magnetic Fluid Based Squeeze Film behavior between curved circular Plates and Surface Roughness Effect Nikhilkumar D. Abhangi 1 and G. M. Deheri 1 Abstract: Efforts have been directed to study and analyze the behavior of a magnetic-fluid-based squeeze film between curved rough circular plates when the curved upper plate (with surface determined by an exponential expression) approaches the stationary curved lower plate (with surface governed by a secant function). A magnetic fluid is used as the lubricant in the presence of an external magnetic field oblique to the radial axis. The bearing surfaces are assumed to be transversely rough and the related roughness is characterized via a stochastic random variable with non-zero mean variance and skewness. The associated Reynolds equation is averaged with to the random roughness parameter; then the related non-dimensional differential equation is solved with suitable boundary conditions in dimensionless form to obtain the pressure distribution, such a distribution being necessary for determining the expression of load carrying capacity and ensuing calculation of the response time. The results, presented graphically, indicate that the bearing system displays considerably improved performances as compared to bearing systems working with conventional lubricants. It is seen that the pressure, load carrying capacity and the response time increase with increasing the magnetization parameter. In particular, the load carrying capacity increases with to the upper plate s curvature parameter, while a symmetric distribution takes place with regard to the lower plate s curvature parameter. Even if the effect of transverse roughness is adverse in general, this investigation offers some indications for obtaining better performance in the case of negatively skewed roughness (by suitably choosing the curvature parameters of both the plates). Keywords: Magnetic Fluid, Squeeze film, Transverse roughness, Reynolds equation, Load carrying capacity. Nomenclature a Radius of the circular plate 1 Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar , Gujarat, India

2 246 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 p B H P Lubricant pressure urvature parameter of the upper plate urvature parameter of the lower plate Magnitude of the magnetic field Dimensionless pressure (P = h3 0 p µḣ 0 a 2 ) Load carrying capacity Dimensionless load carrying capacity t Response time T Non-dimensional response time α Mean of the stochastic film thickness σ Standard deviation of the stochastic film thickness ε Measure of symmetry of the stochastic random variable σ σ/h 0 α α/h 0 ε ε/h 3 0 φ Inclination angle µ Absolute viscosity of the lubricant µ Magnetic susceptibility µ 0 Permeability of the free space µ Magnetization parameter (µ = µ 0µkh 3 ) µḣ 0 1 Introduction The behavior of a squeeze film between various geometrical configurations of flat surfaces was analyzed by Archibald (1956). Murti (1974) discussed the performance of a squeeze film trapped between curved circular plates describing the film thickness by an exponential expression and it was established that the load carrying capacity increased sharply with curvature in the case of concave pads. Gupta and Vora (1980) investigated the corresponding problem considering the annular plates. Here the lower plate was taken to be flat. Ajwaliya (1984) studied this problem of squeeze film behavior taking the lower plate also to be curved. u (1970), (1972) dealt with the squeeze film performance when one of the surface was porous faced taking mainly two types of geometries namely, annular and rectangular. Various bearing configurations such as circular, annular, elliptical, rectangular and conical were investigated by Prakash and Vij (1973). They made a comparison between the squeeze film performances of different geometries of equivalent surface area. It was concluded that the circular plates registered highest transient load carrying capacity, other parameters remaining same.

3 Magnetic Fluid Based Squeeze Film behavior 247 The above studies made use of conventional lubricants. The application of a magnetic fluid as a lubricant was analyzed by Verma (1986). The magnetic fluid comprised of fine surfactant and magnetically passive solvent. Subsequently, the squeeze film behavior between porous annular disks in the presence of a magnetic fluid lubricant was presented by Bhat and Deheri (1991). It was established that the application of magnetic fluid lubricant enhanced the performance of the squeeze film. However, the plates were considered to be flat. But, in actual practice the flatness of the plate does not endure owing to elastic, thermal and uneven wear effects. ith this end in view Bhat and Deheri (1993) studied the behavior of a magnetic fluid based squeeze film between cured circular plates. The magnetic fluid based squeeze film between curved plates lying along the surfaces determined by exponential, secant and hyperbolic function was analyzed by Patel and Deheri (2008), (2002.a), (2002.b). It was established that the application of magnetic fluid lubricant improved the performance of the squeeze film. It is a well-established fact that the bearing surfaces tend to develop roughness after having some run-in and wear. The roughness appears to be random and disordered. The randomness and the multiple roughness scales both contribute to be complexity of the geometrical structure of the surfaces. Invariably, it is this complexity which contributes to most of the problems in studying friction and wear. The random character of the surface roughness was recognized by several investigators who resorted to a stochastic approach in order to mathematically model the roughness of the bearing surfaces (Tzeng and Seibel (1967), hristensen and Tonder (1969.a), (1969.b), (1970)). Tonder (1972) analyzed theoretically the transition between surface distributed waviness and random roughness. Tzeng and Seibel (1967) dealt with a beta probability density function for the random variable characterizing the roughness. This distribution is symmetrical in nature with zero mean and approximates the Gaussian distribution to a good degree of accuracy for certain special situations. hristensen and Tonder (1969.a), (1969.b), (1970) developed and modified this approach of Tzeng and Seibel (1967) in order to propose a comprehensive general analysis both for transverse as well as longitudinal surface roughness based on a general probability density function. The method adopted by hristensen and Tonder (1969.a), (1969.b), (1970) laid the frame work to analyze the effect of surface roughness on the performance of a bearing system in a number of investigations (Ting (1975), Prakash and Tiwari (1983), Prajapati (1991), Guha (1993), Gupta and Deheri (1996)). In most of these analyses the probability density function for the random variable characterizing the surface roughness was assumed to be symmetric with mean of the random variable equal to zero. However, in general this may only be true to the first approximation. In practice due to non-uniform rubbing of the surfaces the distribution of surface roughness may indeed be asym-

4 248 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 metrical. ith this idea in view, Andharia, Gupta and Deheri (1997) discussed the effect of transverse surface roughness on the performance of a hydrodynamic squeeze film in a spherical bearing making use of general stochastic analysis. It was observed that the effect of transverse surface roughness on the performance of the bearing system turned out to be considerably adverse. Here it has been proposed to study and analyze a magnetic fluid based squeeze film between curved transversely rough circular plates where in, the upper plate lies along the surface determined by an exponential expression while the lower plate is taken along a surface governed by secant function. 2 Analysis Fig. 1 shows the configuration of the bearing system. Figure 1: Bearing onfiguration The bearing surfaces are assumed to be transversely rough. The thickness h(x) of the lubricant film is h(x) = h(x) + h s (1) where h(x) is the mean film thickness while h s is the deviation form the mean film thickness characterizing the random roughness of the bearing surfaces. The deviation h s is considered to be stochastic in nature and governed by the probability density function f (h s ), c h s c (2) where c is the maximum deviation from the mean film thickness. The mean α, the standard deviation σ and the parameter ε which is the measure of symmetry associated with random variable h s are governed by the relations α = E(h s ), (3)

5 Magnetic Fluid Based Squeeze Film behavior 249 σ 2 = E[(h s α) 2 ] and ε = E[(h s α) 3 ] (4) (5) where E denotes the expected value defined by c E(R) = c R f (h s )dh s (6) It is taken into consideration that the upper plate lying along the surface determined by Z u = h 0 [ exp( Br 2 ) ] ; 0 r a (7) approaches with normal velocity ḣ 0 = dh 0 dt, to the lower plate lying along the surface Z l = h 0 [sec( r 2 ) 1]; 0 r a (8) where h 0 the central distance between the plates and B and are the curvature parameters of the corresponding plates. The central film thickness h(r) then is defined by h(r) = h 0 [ exp( Br 2 ) sec( r 2 ) + 1 ] (9) Axially symmetric flow of the magnetic fluid between the plates is taken into account under an oblique magnetic field H = (H(r)cosφ(r,z), 0, H(r)sinφ(r,z)) (10) whose magnitude H vanishes at r = a; for instance; H 2 = ka(a r), 0 r a where k is a suitably chosen constant so as to have a magnetic field of required strength, which suits the dimensions of both the sides. The direction of the magnetic field plays a pivotal role since H has to satisfy the equation H = 0, H = 0. (11) Therefore, H arises out of a potential function and the inclination angle φ of the magnetic field H with the lower plate is determined by the first order partial differential equation cotφ φ r + φ z = 1 2(a r) (12)

6 250 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 whose solution is determined from the equations c 2 1 cosec 2 φ = a r, z = 2c 1 (a c 2 1 r ) (13) where c 1 is a constant of integration. The modified Reynolds equation governing the film pressure p then can be obtained as [(1996), (2002.a), (2002.b)] [ 1 d rg(h) d ( p 0.5µ0 µh 2)] = 12µḣ 0 (14) r dr dr where g(h) = h 3 + 3σ 2 h + 3h 2 α + 3hα 2 + 3σ 2 α + α 3 + ε (15) Introducing the non-dimensional quantities h = h/h 0, R = r/a, µ = µ 0µkh 3 µḣ 0, P = h3 0 p µa 2ḣ 0 (16) σ = σ/h 0, ε = ε/h 3 0, B = Ba, = a 2 and solving the concerned Reynolds equation with the associated boundary conditions P(1) = 0, dp dr = µ at R = 0 (17) 2 one can avail the non-dimensional pressure distribution as P = µ 2 where in 1 (1 R) + 6 R R dr (18) G(h) G(h) = h 3 + 3h 2 α +3σ 2 h + 3hα 2 +ε +3σ 2 α + 3 The dimensionless load carrying capacity is given by = h3 0 2πµa 4ḣ = µ R 3 dr (19) G(h)

7 Magnetic Fluid Based Squeeze Film behavior 251 where the load carrying capacity is obtained from the relation a = 2π rp(r)dr 0 (20) Lastly, the response time in dimensionless form is determined from the relation T = th2 0 πµa 4 = h 2 h 1 1 dh (21) G(h) where h 1 = h 1 h 0, h 2 = h 2 h 0 (22) 3 Results and discussions Equations (18), (19) and (21) represent the expressions for non-dimensional pressure P, load carrying capacity and response time T. It is evident that these performance characteristics depend on various parameters such as µ,,,, B and. These parameters describe ively, the effect of magnetic fluid lubricant, standard deviation of roughness, variance associated with roughness, measure of symmetry, the upper plate s curvature parameter and the lower plate s curvature parameter. The equation (19) tends to suggest that the load carrying capacity of the bearing increases by µ. Setting the roughness parameters, and to be zero one gets the performance of a magnetic fluid based squeeze film trapped between curved circular plates lying along the surfaces determined by exponential function and secant function. Furthermore, taking the magnetization parameter to be zero the present study reduces to the performance of squeeze film behavior between curved circular plates. The variation of load carrying capacity with to the magnetization parameter µ is presented for various values of roughness parameters,, and the curvature parameters B and ively in Fig It is indicated from these figures that the load carrying capacity rises sharply with to the magnetization parameter although, the effect of µ is almost negligible up to the value 0.01 as shown in Fig Besides, among the roughness parameters the combined effect of the magnetization parameter and skewness is more pronounced.

8 to suggest that the earing increases by ghness parameters ets the performance ueeze film trapped s lying along the ntial function and re, taking the ro to the suggest present that study the earing ueeze film increases behavior by ghness parameters arrying ets the performance capacity ueeze n parameter film trapped μ * is s lying along the ughness parameters ntial function and arameters B and re, taking the ro the present study ueeze film behavior arrying capacity n parameter μ * is ughness parameters arameters B and = rying capacity with = rying capacity with =0 rying capacity with =0 rying capacity with σ *, α * and ε * to be zero one gets the performance 1.12 of a magnetic fluid based squeeze film trapped between curved circular plates lying along the surfaces determined by exponential function and secant function. Furthermore, taking the 0.92 magnetization parameter to be zero the present study = = = reduces to the performance of squeeze film behavior =0.01 =0.02 between curved circular plates. The variation of load carrying capacity Figure 4: Variation = of load 0 carrying =0.05 capacity with = 0.1 with to the magnetization parameter μ * is to μ * and ε * =0.15 =0.2 presented for various values of roughness parameters σ *, α *, ε * and the curvature parameters B and Figure 2: Variation of load carrying capacity with ively in Fig to μ * and σ * The equation (19) tends 0.6 to 0.8suggest 1 that the B=-0.2 B=-0.1 B=0 load carrying capacity of the bearing increases by μ *. Setting the roughness parameters 1.11 Figure 5: Variation of load carrying capacity with = 0 =0.05 = 0.1 σ *, α * and ε * to be zero one gets the performance =0.15 =0.2 to μ * and B = 0.05 = =0 =0.025 =0.05 of a magnetic fluid based squeeze film trapped 1.02 Figure between curved of circular load carrying plates capacity lying along with the Figure 3: of load carrying capacity with 2: to Variation μ * and σ * of load carrying capacity secant withfunction. to Furthermore, µ and taking the pacity with to Figure 0.99 surfaces determined by exponential function and 3: Variation of load carrying ca to 0μ * and 0.2α * µ and magnetization 1.11 parameter to be zero the present study reduces = 0.02 = 0.01 =0 to the performance of squeeze film behavior between 1.02 =0.01 =0.02 curved circular plates The variation of load carrying capacity 0.99 Figure 4: Variation of load carrying capacity with with 0 0.2to the 0.4magnetization parameter 1 μ * is to μ * and ε * =-0.2 =-0.1 =0 presented for various values of roughness parameters =0.1 = σ *, = 0.02 α *, ε * and the = 0.01 curvature parameters =0 B and ively Figure 6: Variation of load carrying capacity with 1.11 =0.01 = 0.05 in Fig =0.02 = =0 =0.025 =0.05 to μ * and 0.87 Figure 4: Variation of load carrying capacity with 0.78 Figure 3: Variation of load carrying capacity with 1.02 to μ1.12 * and ε * It is indicated 0.69from these figures that the load to μ * and α * carrying capacity rises sharply with to the magnetization 0.51 parameter although, the effect of μ * 0.97 is almost negligible 0 up to 0.2 the value as 0.6 shown 0.8 in = 0.02 = 0.01 =0 Fig =0.01 = : Variation of load carrying capacity with 5: Variation of load carrying capacity with Figure 0.514: Variation = 0 of load =0.05 carrying = capacity with to µ 0.6 and0.8 1 pacity with to µ and B 0.1 Figure 5: Variation of load carrying ca- to μ * and =0.15 ε * =0.2 to μ * and B Figure 2: Variation of load carrying capacity with to B=0.1 μ * and σ * B= Figure : Variation 1.15 of load carrying capacity with 0.69 to μ * and B =-0.2 =-0.1 = =0.1 = Figure : Variation of load carrying capacity with Figure 6: Variation of load carrying capacity with to μ * and = 0.05 B = = = = to μ * and Figure 3: Variation of load carrying capacity with It is indicated from these figures that the load =-0.2 =-0.1 =0 = =0.001 =0.01 to =0.1 μ * and α * =0.2 carrying capacity =0.1 rises sharply =1 with to the 1.11 magnetization parameter although, the effect of μ * 6: Variation of load carrying capacity with is almost 7: negligible up of to load the value carrying 0.01 capacity as shown with Figure 6: Variation of load carrying capacity 1.02with to µ and pacity with to and µ Figure 7: Variation of load carrying ca- in to μ * and Fig to σ * and μ * 1.15 It is indicated 0 from 0.2 these 0.4 figures 0.6 that 0.8 the 1load carrying capacity rises sharply with to the magnetization =-0.2 parameter although, =-0.1 the effect =0 of μ * is almost negligible =0.1 up to the =0.2 value 0.01 as shown in Fig Figure 6: Variation of load carrying capacity with to μ * and 252 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 It is indicated from these figures that the load carrying capacity rises sharply with to the magnetization parameter although, the effect of μ * = =0.001 =0.01 =0.1 =1 Figure 8: Variation of load carrying capacity with B=-0.2 B=0.1 Figure 5: Variation o to μ * and B =-0.2 =0.1 Figure 6: Variation o to μ * and It is indicated from carrying capacity rises magnetization paramet is almost negligible up Fig = =0.1 Figure 11: Variation o to and μ * = 0.05 =0.025 Figure 12: Variation o

9 1.12 = =0.001 =0.01 = =0.001 = =0.1 =1 = = Figure 7: Variation of load carrying capacity with Figure 11: Variation of 0.97 load carrying capacity with to σ * and μ * to and μ * Magnetic Fluid Based = =0.001 Squeeze Film =0.01 behavior = =0.001 = = =0.001 =0.01 = 0.05 =0.1 =1 =0.1 =0.1 =1 =1 =0.025 Figure 7: Variation of load carrying capacity with Figure Figure 11: Variation 8: Variation of load of carrying load carrying capacity capacity with with Figure 12: Variation o to σ * and μ * to 0 and to μ α0.05 * and μ * to σ * and α * = =0.001 =0.01 = 0.05 = =0 =0.1 =1 =0.025 = Figure 8: Variation of load carrying capacity with Figure 12: Variation of load carrying capacity with to α * and μ * to σ * and α * = =0.001 =0.01 = 0.05 = =0 = =0.001 =0.01 = 0.02 =0.1 =1 =0.025 =0.1 =0.05 =1 =0.01 8: Variation of 1.02 load carrying capacity with Figure 12: Variation of load carrying capacity with Figure to 8: α * Variation and μ * of load carrying ca- 9: of load carrying capacity with Figure 13: Variation o Figure to σ * to and 9: Variation ε * α and * of load carrying capacity with μ * to σ * and ε * pacity-0.02 with to 0 and 0.01 µ 0.02 to and µ 1.12 = =0.001 =0.01 = 0.02 = 0.01 = = =1 = = Figure 9: Variation 0.97 of load carrying capacity with Figure 13: Variation of load carrying capacity with to ε * and μ * 1.02 to σ * and ε * = B =0.001 =0.01 = =0.001 =0.01 = =0.001 =0.01 = 0.02 = 0.01 =0 B= = =1 =0.1 =0.1 =1 =0.01 = =0.02 =0.001 =1 =0.01 B= =0.1 =1 Figure 7: Variation 0.69 of load carrying capacity with 0.55 Figure 11: Variation of 0.97 load carrying capacity with Figure 9: Variation of load carrying capacity with Figure 13: Variation of load carrying capacity with Figure 14: Variation o to σ * and 0.60μ * 0.45 Figure 10: to Variation and μ * of 1.02 load carrying capacity with to ε * 0.92 and μ * 0.51 to σ * and ε * to σ * and B 0 to B 0.05 and μ * B = = =0.001 =0.001 =0.01 =0.01 = =0.001 =0.01 =0.1 =0.1 =1 =1 =0.1 = Figure 14: Variation of load carrying capacity with Figure 10: Variation 7: Variation of load of carrying load carrying capacity capacity with with Figure 11: Variation of load carrying capacity with 0.55 Figure to σ * and B to 10: B and to Variation μ σ * and 0.60μ * of load carrying Figure 0.45 to 11: and Variation μ * of load carrying capacity with to B and µ capacity with to and µ =0.01 = = =0.001 B =0.001 =0.01 =0.01 = 0.05 = =0 =0.1 = =0.001 =0.1 =1 =1 =0.01 B=0.1 =0.025 B=0.2 =0.05 =0.1 =1 arrying capacity with Figure Figure 11: Variation 8: Variation of load of carrying load carrying capacity capacity with Figure with Figure 14: Variation 12: Variation of load of carrying load carrying capacity capacity with with Figure 10: to Variation and to μ α * and of load μ * carrying capacity with to σ * and to σ B * and α * to B and μ * = =0.001 =0.01 = 0.05 = =0 =0.1 =1 =0.025 =0.05 Figure 8: Variation of 1.02 load carrying capacity with Figure 12: Variation of load carrying capacity with 0 to α0.05 * and μ * to σ * and α * =0.01 = 0.05 = =0 = =0.001 =0.01 = 0.02 = 0.01 =0 =0.025 = =0.1 =1 =0.01 =0.02 arrying capacity with 12: Variation of load carrying capacity with Figure Figure to 12: 9: σ * and Variation of load α * of carrying load carrying capacity with 13: Variation of load carrying capacity with Figure to ε * and μ * 1.02 to 13: σ * and Variation ε * of load carrying capacity with to 0.99 and capacity 0with to 0.15 and = =0.001 =0.01 = 0.02 = 0.01 = =0.1 =1 =0.01 = Figure 9: Variation of load carrying capacity with Figure : Variation of load carrying capacity with to ε * and μ * to σ * and ε * B.001 =0.01 = 0.02 = 0.01 =0 =0.01 = =0.02 =0.001 =0.01 = = rying capacity with Figure 13: Variation of load carrying capacity with Figure 14: Variation of load carrying capacity with Figure 10: Variation of 0.69 load carrying capacity with 0.55 to σ * and B

10 =0.01 rying capacity with = 0.02 = 0.01 =0 =0.01 = opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 Figure 13: Variation of load carrying capacity with to σ * and ε * =0.01 arrying capacity with Figure 14: Variation of load carrying capacity with to σ * and B Figure 14: Variation of load carrying capacity with to and B =0 =0.05 =0.1 =0.15 = =-0.2 =-0.1 =0 0 = = Figure 15: Variation =-0.2 of load =-0.1 carrying capacity =0 with =0.1 =0.2 to σ * and Figure 15: Variation of load carrying capacity with to and Figure 22: Variation of load carrying capacity with = = Figure 19: Variation =0 o =0.15 to B and σ * Figure 15: Variation of load carrying capacity with Figure 19: Variation o to σ * and to B and σ * Fig describes the effect of the standard deviation associated 0.93 with roughness on the distribution of the load carrying capacity It can be easily seen from these figures that the effect of 0.45the standard deviation is =0 =0.05 =0.1 = considerably 0 adverse, in the 0.15sense0.2that the load carrying =0.15 capacity =0.2 decreases substantially, although the effect of standard deviation is negligible upto 0.05 as can be -0.05= B =-0.2 =-0.1 =0 =0 =0.05 =0.1 Figure 16: Variation of load carrying capacity with Figure 20: Variation o =0.1 =0.2 =0.15 =0 =0.2 =0.05 =0.1 = 0.02 =0.15 =0.2 =0.01 seen from Fig to and σ * to α * and ε * Figure 15: Variation of load carrying capacity with Figure Figure 19: Variation 16: Variation of load of 1.11 carrying load carrying capacity capacity with with Figure 20: Variation o to σ * and to B and to σ * and σ * to α * and ε * B= =0 0 = = B= = = =0 =0.05 =0.1 = 0.02 = 0.01 =0 Figure 21: B=-0.2 Variation o =0.15 =0.2 Figure = : =0 Variation =0.02 of load =0.05 carrying capacity =0.1 with B=0.1 =0.15 =0.2 to α * and B to α * and σ * 16: Variation of load carrying capacity with Figure 20: Variation of load carrying capacity with Figure 21: Variation o Figure 16: Variation of load carrying to and σ * Figure 17: to α * and 17: Variation ε * Variation of load carrying of loadcapacity carrying with to α * and B to α * and σ * capacity with 1.11 to and capacity with to and The negative effect of is a little bit less with to the measure of symmetry as compared to that 0.71 of variance associated with roughness = = =0 In Fig , one can have the effect of variance on =0.05 =0.1 =0.15 the variation =0.2 of load carrying B=-0.2 B=-0.1 B=0 Figure 22: Variation =-0.2 capacity. o =0 =0.05 =0.1 =0.15 =0.2 Figure B=0.1 18: Variation B=0.2 =0.1 =0 of load =0.05 carrying capacity =0.1 with to α * and =0.15 =0.2 These figures make it clear that (+ve) to ε * and σ * Figure decreases 21: Variation the load of load carrying carrying capacity capacity with while Figure 22: Variation o Figure 17: Variation of load carrying capacity with (-ve) increases the load carrying capacity. Figure to Furthermore, α * and 18: B Variation of itload is indicated carrying capacity thatwith the to α * and to α * and σ * to ε * and σ * combined effect of the upper plate s curvature parameter and the negative variance tends to be significantly positive. The effect of the measure of symmetry on the =-0.2 =-0.1 =0 =0.1 =0.2

11 Magnetic Fluid Based Squeeze Film behavior B=-0.2 B=-0.1 B=0 255 =0 =0.05 =0.1 =0.15 =0.2 Figure 21: Variation of load carrying capacity with Figure 17: Variation 0.93 of load carrying capacity with to α * and B 0.55 to α * and σ * B 0.93 =-0.2 =-0.1 =0 =0 =0.05 = =0.1 =0.2 =0.15 = Figure : Variation of load carrying capacity with 0.00 Figure : Variation of load carrying capacity with 0.55 to σ * and B to B and σ * =-0.2 =-0.1 =0 =0 =0.05 = =0.1 = = =0.2 =-0.2 =-0.1 B =0 =0.1 =0.2 =0 =-0.2 =0.05 =-0.1=0.1 =0 =0 =0.05 =0.1 re 15: Variation of load carrying =0.15 capacity with Figure 19: Variation of load carrying capacity =0.1 =0.2 =0.2 =0.15with =0.2 ct to σ * and to B and σ * Figure 22: Variation of load carrying capacity with Figure 18: Variation 15: Variation of 0.55 load of carrying 0.93 load carrying capacity capacity with with Figure to α * and 19: Variation of load carrying capacity with Figure 18: of load carrying to ε * and to σ * and 0.45 to B and σ * capacity with to 0.10 and 0.20 capacity with to B and B =0 =0 =0.05 = =0.15=0 =0.2 =0.05 =0.1 = 0.02 = 0.01 = =0.15 =0.2 =0.01 = =0.1.2 arrying capacity with Figure 22: Variation of load carrying capacity with to α * and Figure 19: Variation of load carrying arrying capacity with Figure 19: Variation of load carrying capacity with Figure 16: Variation of load carrying capacity with Figure 20: Variation of load carrying capacity with to 0.1 B and σ * to and σ * to α * and 0.05 ε * =0 =0.05 =0.1 = 0.02 = 0.01 = =0.15 =0.2 =0.01 = =0 =0.05 =0.1 = 0.02 = 0.01 =0 =0.15 =0.2 = = re 16: Variation of load carrying capacity with Figure 20: Variation of load carrying capacity with ct to and σ * to α * and ε * Figure 16: Variation of load carrying capacity with Figure 20: Variation of load carrying capacity with to and 0.70 σ * 0.71 to α * and ε * =0.1 = 0.02 = = =0.01 =0 =0.02 =0.05 = B= B= =0.15 = arrying capacity with 20: Variation of load 0.81 carrying capacity with 21: Variation of load carrying capacity with 0.71Figure Figure to 20: α * and 17: Variation ε * of ofload load carrying carrying capacity 0.60 with Figure 21: Variation of load carrying 0.71 to α * and B 0.60 capacity with to α * and σ * to and capacity with to and B =0 =0.05 = =0.15 =0.2 = =0.05 =0.1 = =0.2 Figure 21: Variation of load carrying capacity with re 17: Variation of load carrying capacity with Figure 21: Variation of load carrying capacity with Figure 17: Variation 0.60 of load carrying to α * and capacity B with to α * and B 0.70 ct to α * and σ * to α * and σ * B=-0.2 B=-0.1 B=0 =-0.2 =-0.1 =0.05 =0.1 =0.1.2 =0 =0.05 =0.1 =0.2 =0.15 =0.2 Figure 21: Variation of load carrying capacity with Figure 22: Variation of load carrying capacity with arrying capacity with Figure to α * and 18: B Variation of load carrying capacity 0.70 with to α * and 0.70 to ε * and σ * =-0.2 =-0.1 =0 =-0.2 =-0.1 =0 =0.1 =0.2 =0 =0.05 =0.1 =0.1 =0.2 =0 =0.05 =0.1 =0.15 =0.2 =0.15 =0.2 Figure 22: Variation of load carrying capacity with 18: 0.70 Figure 22: Variation of load carrying capacity with of load carrying capacity with re 18: Variation of load carrying Figure capacity 22: with Variation to α * and to ε * and σ of * to load α * and carrying capacity with to and ct to ε * and σ * =-0.2 =-0.1 =0 =0.1 =0.2

12 256 opyright 2009 Tech Science Press FDMP, vol.5, no.3, 0.00 pp , distribution of load carrying capacity is depicted in Fig Figure 23: Variation of load carrying capacity with to ε * and B Figure 27: Variation o to and ε * 23: Variation of load carrying capacity with Figure 27: Variation 24: Variation of load of carrying load carrying capacity capacity with with Figure 23: Variation of load carrying Figure to ε * and B to and to 24: ε * and Variation of load carrying In addition, th capacity with to and B capacity with to and the negatively skewe plate s curvature para effect of the negative lower plate s curvature As in the case of variance here also the load carrying capacity decreases due to noticed that the rate 0.93 capacity with t positively skewed roughness, while the negatively skewed 0.55 roughness increases the is more with t load carrying capacity In 0.01 addition, 0.02there is the symmetric 0.45 distribution of the load parameter as compar curvature parameter. B carrying capacity with to the lower plate s curvature =-0.2 =-0.1 =0 parameter which can =0 =0.05 =0.1 positive effect induc =0.1 =0.2 be seen from Fig =0.15 =0.2 roughness gets further variance resulting in th In Figure addition, 24: Variation the of combined load carrying positive capacity with effect offigure the negatively 25: Variation skewed of load carrying roughness capacity with and of negatively skewe to ε * and In addition, to and the σ combined * variance is significant the upper plate s curvature parameter dominates the positive effect positive of effect theof negatively skewed roughness and the lower plate s curvature parameter parameter. dominates Interestingly, the positive it expression (21) it is fou the negatively skewed roughness and the upper time Δ T are identical capacity. A comparison is noticed that the rate of increase in loadeffect carrying of the negatively capacityskewed with roughness and lower plate s curvature parameter. to the magnetization parameter is more with noticed to the that lower the rate plate s of increase curvature in load carrying parameter study of Patel and D Interestingly, it is increase in load carry more considerable here 0.93 capacity with to the magnetization parameter as compared to that of upper plate s curvature parameter. Besides, 0.70 it is revealed is more with to the lower plate s curvature 4 onclusion that the positive effect induced by negatively parameter skewed as compared roughness to that of upper plate s gets further enhanced owing to negative variance resulting roughness is adverse, Albeit, in ge curvature parameter. Besides, it is revealed that the =0 =0.05 =0.1 positive ineffect the fact = 0.05 induced thatby the= negatively combined skewed =0 effect of properly choosing the negatively=0.15 skewed roughness =0.2 and negative roughness variance gets further is =0.025 significantly enhanced =0.05 owing positive. to negative Lastly, the plates and the variance resulting in the fact that the combined effect from Figure the 25: expression Variation of load (21) carrying it iscapacity foundwith that of thenegatively trends Figure of 26: skewed response Variation roughness of time load carrying T and are negative capacity identical with performance of the bea to and σ * variance significantly to and α * considerably in the with those of load carrying capacity. A comparison of this positive. investigation Lastly, from with the the expression (21) it is found that the trends of response study of Patel and Deheri [12] suggeststime thatδ Tthe are increase identical with in load those of carrying load carrying capacity is substantially capacity. A comparison of this investigation with the more considerable here. study of Patel and Deheri [12] suggests that the increase in load carrying capacity is substantially more considerable here. 4 onclusion onclusion Albeit, in general, the effect of transverse roughness Albeit, in general, is adverse, the effect of this transverse article reveals that by properly choosing the curvature = 0.05 = =0 roughness is adverse, this article reveals that by =0.025 =0.05 properly choosing parameter the curvature of bothparameter the plates of both and the magnetization parameter the performance the plates of theand bearing the magnetization system can parameter be improved the Figure 26: Variation of load carrying capacity with performance of the bearing system can be improved considerably to and α in * the case of negatively skewed considerably roughness. in the case This of negatively investigation skewed offers = 0.02 =-0.2 = 0.01 =-0.1=0 =0 =0.01 =0.1 =0.02 =0.2 some indications even for extending the life period of the bearing system = 0.02 = B=-0.2 B=0.1

13 B= =-0.2 =-0.1 =0 =0 =0.05 =0.1 =0.1 =0.2 B=-0.2 =0.15 B=-0.1 =0.2 B=0 Figure 24: Variation of load carrying capacity with Figure 25: Variation of load carrying capacity with to ε * and In addition, to and the σ combined * positive effect of the negatively skewed roughness and the upper plate s curvature parameter dominates the positive effect of the negatively skewed roughness and the lower plate s curvature parameter. Interestingly, it is 1.07 noticed that the rate of increase 1.07 in load carrying capacity with to the magnetization parameter is more with to the lower plate s curvature parameter as compared to that of upper plate s 0.57 curvature parameter. Besides, 0.99 it is revealed that the =0 = positive effect = 0.05 induced by 0.97 = negatively skewed =0 =0.1 = = roughness gets further = enhanced = owing to 0.10 negative 0.20 variance resulting in the fact that the combined effect 25: Variation of load carrying capacity with of negatively 26: skewed Variation roughness of load carrying and negative capacity with Figure 25: Variation B=-0.2 of load B=-0.1 carrying B=0 Figure to and σ * variance is significantly to 26: = 0.02 and Variation = 0.01 α * of load=0 carrying =0.01positive. =0.02 Lastly, from the capacity with to and expression capacity (21) is with found that the trends to of and response Figure 23: Variation of load carrying capacity time with Δ TFigure are identical 27: Variation with of those load of carrying load carrying capacity with to ε * and B capacity. A comparison to and of ε * this investigation with the study of Patel and Deheri [12] suggests that the increase in load carrying capacity is substantially 1.07 more considerable here onclusion 0.99 Albeit, in general, the effect of transverse = = =0 roughness is adverse, this article 0.55 reveals that by = = properly choosing the curvature parameter of both the plates and the magnetization parameter the Figure 26: Variation of load carrying capacity with performance of the bearing system can be improved = 0.02 =-0.2 = 0.01 =-0.1=0 =0 to and α * considerably in the case of negatively skewed =0.01 =0.1 =0.02 =0.2 Magnetic Fluid Based Squeeze Film behavior 257 arrying capacity with Figure 27: Variation 24: Variation of load of carrying load carrying capacity capacity with with Figure 27: to and to ε * and of load carrying Figure 28: Variation of load carrying In addition, the combined positive effect of capacity with to and capacity the negatively withskewed roughness to and the upper plate s curvature parameter dominates the positive effect of the negatively skewed roughness and the lower plate s curvature parameter. Interestingly, it is noticed that the rate of increase in load carrying References 0.93 capacity with to the magnetization parameter 0.55 is more with to the lower plate s curvature parameter compared to that of upper plate s Ajwaliya, -0.2 M. B. (1984): -0.1 On 0 certain 0.1 theoretical 0.2 aspects of lubrication, Ph. D. thesis, curvature parameter. Besides, it is revealed that the -0.1 =0 Sardar Patel University, =0 Vallabh =0.05 Vidyanagar. =0.1 positive effect induced by negatively skewed 0.2 B=-0.2 =0.15 B=-0.1 =0.2 B=0 roughness gets further enhanced owing to negative Andharia, P. B=0.1 I., Gupta, B=0.2 J. L., Deheri, G. M. variance (1997): resulting Effect in the offact longitudinal that the combined surface effect arrying capacity with Figure 25: Variation of load carrying capacity with of negatively skewed roughness and negative roughness on hydrodynamic lubrication of slider bearings. Proceedings of Tenth addition, to and the σ combined * variance is significantly positive. Lastly, from the positive effect of International onference on Surface Modification expression (21) Technologies, it is found that the trends Institute of response of the negatively skewed roughness and the upper time Δ T are identical with those of load carrying Materials, plate s curvature Singapore, parameter dominates pp the positive capacity. A comparison of this investigation with the effect of the negatively skewed roughness and the study of Patel and Deheri [12] suggests that the Archibald, lower plate s curvature F. R. (1956): parameter. Interestingly, Load capacity is and time relations for squeeze films, Jour. increase in load carrying capacity is substantially noticed that the rate of increase in load carrying Basic Engg. Trans. ASME. Sear. D, Vol.78, pp more considerable here. capacity with to the magnetization 0.70 parameter is more with to the lower plate s curvature Bhat, M. V., -0.2 Deheri, -0.1 G. M. 0 (1991): 0.1 Squeeze film onclusion behavior in porous annular disks parameter as compared to that of upper plate s Albeit, in general, the effect of transverse lubricated curvature parameter. with magnetic Besides, it is fluid, revealed EAR, that the Vol. 151, pp roughness is adverse, this article reveals that by.05 =0.1 positive effect = 0.05 induced by = negatively skewed =0 properly choosing the curvature parameter of both.2 Bhat, roughness M. gets V., further =0.025 Deheri, enhanced G. =0.05 owing M. (1993): to negative Magnetic fluid based squeeze film between the plates and the magnetization parameter the variance resulting in the fact that the combined effect porous arrying capacity with of negatively Figure circular 26: skewed Variation disks, roughness of J. load Indian carrying Acad. and negative capacity Math., with performance Vol. 15(2), of the pp bearing system can be improved variance significantly to and α * considerably in the case of negatively skewed hristensen, H., Tonder, positive. Lastly, K.. from (1969.a): the Tribology of rough surfaces: stochastic expression (21) it is found that the trends of response models time Δ T of are hydrodynamic identical with those lubrication., of load carrying SINTEF report no.10/ capacity. A comparison of this investigation with the study of Patel and Deheri [12] suggests that the increase in load carrying capacity is substantially more considerable here. noticed that the rate capacity with t is more with t parameter as compar curvature parameter. B positive effect induc roughness gets further variance resulting in th of negatively skewe variance is significant expression (21) it is fou time Δ T are identical capacity. A comparison study of Patel and D increase in load carry more considerable here 4 onclusion Albeit, in ge roughness is adverse, properly choosing the the plates and the performance of the bea considerably in the = onclusion Albeit, in general, the effect of transverse roughness is adverse, this article reveals that by properly choosing the curvature parameter of both the plates and the magnetization parameter the

14 258 opyright 2009 Tech Science Press FDMP, vol.5, no.3, pp , 2009 hristensen, H., Tonder, K.. (1969.b): Tribology of rough surfaces: parametric study and comparison of lubrication models.,sintef report no. 22/ hristensen, H., Tonder, K.. (1970): The hydrodynamic lubrication of rough bearing surfaces of finite width.asme-asle Lubrication onference, incinnati, Ohio, paper no. 70-Lub-7. Guha, S. K. (1993): Analysis of dynamic characteristics of hydrodynamic journal bearings with isotropic roughness effects. EAR, Vol. 167, pp Gupta, J. L., Deheri, G. M. (1996): Effect of roughness on the behavior of squeeze film in a spherical bearing. Tribol. Trans., Vol.39, pp Gupta, J. L., Vora K. H. (1980): Analysis of squeeze films between curved annular plates, J. Lub. Tech. Trans. ASME, Vol.102.JAN pp.48. Murti, P. R. K. (1974): Squeeze film behavior in porous circular disks, J. Lub. Tech.,Trans. ASME., F, Vol. 95, pp Patel, R. M., Deheri, G. M. (2002.a): Magnetic fluid based squeeze film between two curved plates lying along the surfaces determined by secant functions. Indian Journal of Engineering & Material Sciences, 9, pp Patel, R. M., Deheri, G. M. (2002.b): An analysis of magnetic fluid based squeeze film between curved plates. Journal Nat.Acad. Math., Vol.16, pp Patel, R. M., Deheri, G. M., Patel H.. (2008): Squeeze film behavior in annular plates lubricated with magnetic fluid, To appear in ADIT Journal of Engineering. Prajapati, B. L. (1991): Behavior of squeeze film between rotating porous circular plates: surface roughness and elastic deformation effects. Pure Appl. Math. Sci., Vol.33 (1-2), pp Prakash, J., Tiwari, K. (1983): Roughness effects in porous circular squeezeplates with arbitrary wall thickness. J.Lubr.Technol, Vol. 105, pp Prakash, J., Vij, S. K. (1973): Load capacity and time height relations for squeeze films between porous plates, EAR, Vol. 24, pp Ting, L. L. (1975): Engagement behavior of lubricated porous annular disks. Part1: squeeze film phase, surface roughness and elastic deformation effects. EAR,Vol.34,pp Tonder K.. (1972): Surface distributed waviness and roughness, First world conference in industrial Tribology, New Delhi, A 3, pp.1-8. Tzeng S. T., Saibel E. (1967): Surface roughness effect on slider bearing lubrication, Trans. ASME, J. Lub. Tech., Vol.10, pp Verma, P. D. S. (1986): Magnetic fluid based squeeze film, Int. J. Eng. Sci., Vol.24(3), pp

15 Magnetic Fluid Based Squeeze Film behavior 259 u, H. (1970): Squeeze film behavior for porous annular disks. Trans. ASME, F92, pp u, H. (1972): An analysis of the squeeze film between porous rectangular plates. Trans. ASME, F94, pp

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