International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue, December 8, pp. 98 9, Article ID: IJMET_9 9 Available online at http://www.ia aeme.com/ijmet/issues.asp?jtype=ijmet&vtype= =9&IType= ISSN Print: 976-634 and ISSN Online: 976-6359 IAEME Publication Scopus Indexed STOCHASTICC REYNOLDS EQUATION FOR THE COMBINED EFFECT OF PRESSURE DEPENDENT VISCOSITY AND COUPLE STRESSES ON SQUEEZE-FILM CHARACTERISTICS OF POROUS ANNULAR PLATES Noor jahan and Hanumagowda, B. N. Department of Mathematics, School of Applied Sciences, REVA University, Bangalore, Karnataka, India Sreekala C.K Department of Mathematics, KNS Institute of Technology, Bangalore, Karnataka, India ABSTRACT The paper analyses the effect of piezo viscous dependency on squeeze film characteristics of rough porous annular plates lubricated with couplestress fluid. By using modified Reynold s equation closed form expressions for pressure load supporting capability and squeezing time is derived. The results are presented graphically for Radial and Azimuthal roughness patterns. It is observed that the effect of azimuthal (radial) roughness pattern on the bearing surface is to increase (decrease) the pressure, load carrying capacity and squeeze film time. Due to the change in the pressure dependent viscosity and couple stress parameter the bearing characteristics are increasing and decreases in porous bearings when compared to non-porous bearing. Keywords: Annular Plate, Porous, Rough Surface, Couple stress, Pressure Dependent Viscosity. Cite this Article: Noor jahan and Hanumagowda, B. N and Sreekala C.K, Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates, International Journal of Mechanical Engineering and Technology, 9(), 8, pp. 98 9. http://www.iaeme.com/ijme et/issues.asp?jtype=ijmet&vtype=9&itype e= http://www.iaeme.com/ IJMET/index.asp 98 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K. INTRODUCTION Recently the study about the characteristics of squeeze film containing porous bearing is rapidly increased due to their extended applications in the field of Engineering science especially in improving elasticity, automatic transmissions, lubrication of film elements and artificial joints. Porous bearings are very useful in the field of Engineering due to their selflubricating characteristics and low cost. Due to its wide application many investigators had shown their interest to study the characteristic behaviour of squeeze film with porous condition for different types of bearings. Hai [] studied the behaviour of squeeze film between rotating porous annular disks and observed that the effect of rotating disks reduces the film pressure and load bearing capacity. Naduvinamani et al.[-3] investigated the effect of couple stress on the behaviour of Squeeze film of a short porous journal bearing and rotor bearings and analyzed that the load carrying capacity increases due to the effect of couple stress for both the bearings but co-efficient of friction, decreases for rotor bearings. Rajesh et al. [4] studied the squeezing effect between sphere and flat porous plates and Jaw Ren Lin et al.[5] studied the squeeze film for parallel circular disks lubricated with Ferro fluid. It is observed in both the papers that the characteristics behavior of squeeze film increases when Ferro fluids is used as the lubricant. Murti [6] analysed the squeeze film in full porous metal bearings and concluded that Pressure, Load bearing capacity, and squeeze time is better in long bearings compared to narrow bearings. Bujurke and Patil [7] studied the squeeze film action in porous layered bearings and analysed that due to the influence of elasticity there is an increase in the load bearing capacity and squeezing time. Viscosity in fluids resist the flow of liquid under an applied shear force. Usually it is observed that the viscosity is independent of pressure in solids and gases but in liquids extreme pressure increases viscosity. Usually the effect of viscosity decreases the load bearing capacity; many investigations were carried out to study the effect of viscosity variations. Ayyappa et al. [8] studied the effect of viscosity on the characteristics of squeeze film on short rough journal bearings, Sujatha at el[9] investigated the effects of viscosity on porous parallel rectangular plates and observed that the effect of viscosity decreases the approaching velocity and pressure for short journal bearings and porous parallel rectangular plates. Jaw- Ren Lin et al. [] studied the effect of piezo viscous dependency and couple stress on squeeze film in parallel plate, Martin et al.[] discussed squeeze flow of a peizo viscous fluid, and states that the effect of PDV and non Newtonian couple stress improves the behavior of squeeze film when compared with non-viscous and Newtonian case. Many investigations were carried out to examine effect of surface roughness on different bearings because in industry the surface of the bearing is seen to be rough. In general the surface roughness is of two types that is Azimuthal roughness pattern and radial roughness pattern. Naduvinamani et al. [-3] studied the effects of roughness on hydro magnetic squeeze film between porous rectangular plates and for anisotropic porous rectangular plates, the roughness in the surface increases the characteristic behavior of squeeze films. All the above investigations are carried out to study the behavior of squeeze film and the effects was analyzed by taking different parameter such as couple stress, roughness of surface, porosity, pressure dependent viscosity, on several types of bearings. But so far to best of the authors knowledge the effect of the above discussed parameters on rough porous annular plates have not been discussed, Hence in the present paper, an attempt is made to study the combined effect of pressure dependent viscosity and couple stress on squeeze-film characteristics of rough porous annular plates. http://www.iaeme.com/ IJMET/index.asp 99 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates. MATHEMATICAL FORMULATION The geometrical configuration of rough porous annular plates is shown in figure.the two annular plates approaches each other with normal velocity V (=dh/dt) having film thickness h. The basic equations of motion are given by 4 u u p = 4 dy y r Figure. Physical model of Rough Porous annular plates µ η () p = () y v ( ru) + = r r y (3) The Darcy s law for porous material is k p u = µ ( β ) r (4a) k p v = µ ( β ) y (4b)The components of velocity subject to boundary conditions are At the upper surface y =h; u u =, = y h v = = V (5b) t (5a) At the lower surface y=; u u =, = (6a) y http://www.iaeme.com/ IJMET/index.asp 9 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K v = v (6b) Solution of equation () using the boundary conditions 5(a) and 5(b) is y h Cosh p u y hy l l = + µ r h Cosh l (7) where his film thickness, l ( η µ ) = is couple stress, µ is lubricant viscosity. The relation between viscosity and pressure dependency given by Barus et al. is µ = µ e α p where α denotes the coefficient of pressure-dependent viscosity (PDV) and µ is the viscosity at ambient pressure and a constant temperature. The above relation indicates the lubricant viscosity is increasing exponentially and it could alter the predicted performance of squeeze film bearings. Substituting u from equation (7) in equation (3) and integrating using boundary conditions (6a) and(6b) we get the modified Reynolds Equation as p dh A( h, l, α, p) = µ r r r dt (8) where α p 3 α p α p 3.5α p α p δ ke A( h, l, α, p) = h e l he + 4l e tanh( he / l) + ( β ) (9) The thickness of the fluid film is considered to be made up of two parts in the mathematical model of surface roughness as H = h + h Let f ( h s ) be the probability density function of the stochastic film thickness h s. Taking the stochastic average of modified Reynolds equation (8) with respect to f ( h s ), the stochastic modified Reynolds equation is obtained in the form E { (,,, )} ( p ) dh re A h l α p = µ r r r dt () where E ( ) = ( ) f ( hs ) dhs. For most of the lubricating surfaces, the Gaussian distribution for describing the roughness profile heights is valid up to at least three standard deviations. Following Christensen [4], the roughness distribution function is assumed in the form f ( hs ) = 3c 35 3 7 ( s ). c h c < h < c s elsewhere s http://www.iaeme.com/ IJMET/index.asp 9 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates where c = 3σ and σ is the standard deviation. In the context of Christensen s stochastic theory for the hydrodynamic lubrication of rough surfaces, two types of one dimensional roughness patterns are considered viz., the radial roughness pattern and the azimuthal roughness pattern... Radial Roughness Pattern The one dimensional radial roughness pattern has the form of long, narrow ridges and valleys running in the radial direction (i.e. they are straight ridges and valley H = h + hs ( θ, ξ ), passing through z =, r = to form star pattern), in this case the film thickness takes the form H = h + h ( θ, ξ ) and the average modified Reynolds equation () takes the form s ( ) E p dh E{ A( H, l, α, p) } r = µ r r r dt.. Azimutal Roughness Pattern The one dimensional azimuthal roughness pattern on the bearing surface has the roughness structure in the form of long narrow ridges and valleys running in - direction (i.e. they are circular ridges and valleys on the flat plate that are concentric on z =, r = ). In this case the film thickness assumes the form H = h + h ( r, ξ ), and the averaged modified Reynolds equation (5) takes the form ( ) E p r dh r = µ r r E dt A( H, l, α, p) () Equations () and () together can be written as ( ) E p dh g( H, l, α, p, c) r = µ r r r dt (3) where { α p } { α } E A( H, l,, ) for radial roughness g( H, l, α, p, c) = E / A( H, l,, p) for azimutal roughness s () By introducing non dimensional quantities ( ) αµ b dh dt h l,, G, r, h, H + s, l kδ r h h = ψ = = = = = 3 3 h h h b h h h αµ b 3 h ( dh dt) G =, in equation (3), the Reynolds equation becomes p = µ b E( p) h 3 ( dh dt), p r (,,,,, ) g H l G p ψ C r r r = µ dt dh (4) http://www.iaeme.com/ IJMET/index.asp 9 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K where g H l G p ψ C (,,,,, ) { ψ } { ψ } E f ( H, l, G, p, ) for radial roughness = E / f ( H, l, G, p, ) for azimuthal roughness Gp 3 Gp Gp 3.5Gp.5 Gp ψ e f ( H, l, G, p, ψ ) = e H l e H + 4l e tanh( e H / l ) + ( β ) The non-dimensional Reynolds equation (4) is highly non-linear, so to convert the equation into first order taking the small values of viscosity parameter G = and a perturbation method applied by taking p = p + Gp, we get the following two equations p and p respectively dp r r = r dr g H l ψ (,, ) (,, ψ ) (,, ψ ) g H l d dp d dp r p r = dr dr g dr dr H l (5) (6) where { ψ } { ψ } E f ( H, l, ) for radial roughness g ( H, l, ψ ) = E / f ( H, l, ) for azimuthal roughness { ψ } { ψ } E f ( H, l, ) for radial roughness g ( H, l, ψ ) = E / f ( H, l, ) for azimuthal roughness 3 3 ψ f ( H, l, ψ )) = H l H + 4l tanh( H / l ) + ( β ) (6) 3 3 ψ f ( H, l, ψ )) = H + 6 l H (4 + sec h ( H / l ) 6l tanh( H / l) (7) ( β ) Solving equations (5) and (6) using boundary conditions p = at r =a and p = at r = The dimensionless pressure is obtained as p = g H l a a 3( a ) log r ( r ) (,, ψ ) log ( ) ( log r ) ( log a ) ( 9( a ) g ) ( H, l, ψ ) r + G 3 g( H, l, ψ ) ( a ) (8) The load bearing capacity of the squeeze film is http://www.iaeme.com/ IJMET/index.asp 93 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates W b = π prdr a The non-dimensional load carrying capacity is given as: W ( ) 3 π ( a ) a = + g ( H, l, ψ ) log a a (9) 4 9 π ( a ) g ( 3( ) ( ) ( ) ( H, l, ψ ) a a a + a + G 3 g log ( H, l, ψ ) a ( log a ) 3( a ) where W = µ 3 Wh 4 b ( dh / dt) The non-dimensional squeeze film time is T π ( ) 3 ( a ) a + g ( H, l, ψ ) log a a = dh 4 9 ( ) ( 3( ) ( ) ( ) h π a g ( H, l, ψ ) a a a a + + G 3 g log 3( ) ( H, l, ψ ) a log a a ( ) () http://www.iaeme.com/ IJMET/index.asp 94 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K 3. RESULTS AND DISCUSSION In this paper the squeeze film characteristics of rough porous annular bearing is studied. The discussion is carried out for various non-dimensional quantities such as couple stress parameter l, roughness parameter C and permeability parameter ψ. The result for nondimensional pressure, load bearing capacity, squeezes film time is discussed for roughness patterns such as Azimuthal roughness pattern and Radial roughness pattern. The values for distinct parameter are: h = to, a =.3 to, C = to.4, l = to.4, ψ = to o.. http://www.iaeme.com/ IJMET/index.asp 95 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates 3.. Squeeze Film Pressure In Figure the variation of pressure with different values of C is presented, it is observed that the pressure p is increasing (decreeing) for azimuthal (radial) roughness pattern. In Figure (3) the variation of pressure p against r as function of ψ is depicted and it isobserved that the pressure p is decreasing with increasing values of permeabilityψ. In Figures (4) and (5) the Pressure p with r for distinct values of l and Gis depicted, it is found that pressure increases with increasing values of l and G. http://www.iaeme.com/ IJMET/index.asp 96 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K 3.. Load supporting capacity Figure 6 shows the variation of load carrying capacity W with different values of C, it is observed that the load carrying capacity is increasing (decreasing) for azimuthal (radial) roughness pattern. Figure 7 depicts the load supporting capacity W with respect to h for different values of ψ,and it is observed that the load bearing capacity is decreases with increasing value ofψ. Figure (8) and Figure (9)shows thatthe load supportingcapacity W increases when l and G increases. http://www.iaeme.com/ IJMET/index.asp 97 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates 3.3. Squeeze film time Figure the variation of squeeze film time t with different values of C is presented, it is observed that the squeeze film time is increasing (decreasing) for azimuthal (radial) roughness pattern. In Figure the squeeze film time t with respect to h as a function of ψ is depicted and is observed that squeezing time decreases asψ increases.in Figure and Figures3, the squeeze time t with respect to h as a function of G and l are depicted and it is observed that the squeezing time t increases when the values of l and Gincreases. http://www.iaeme.com/ IJMET/index.asp 98 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K 4. CONCLUSIONS The Stochastic Reynolds Equation for the combined effect of Pressure Dependent Viscosity and Couple Stress on Squeeze-Film Characteristics of porousannular plates is studied in the present analysis and the following results are obtained: The squeeze film Pressure, Load supporting capacity, and squeezing time is increase (decrease) with the increasing value of azimuthal (radial) roughness pattern. The squeeze film Pressure, Load supporting capacity, and squeezing time is decreases with the increase of porous parameter ψ. The squeeze film characteristics such as Pressure, Load supporting capacity, and squeeze film time increases when couple stress l and viscosity parameter G increases. http://www.iaeme.com/ IJMET/index.asp 99 editor@iaeme.com
Stochastic Reynolds Equation For The Combined Effect of Pressure Dependent Viscosity and Couple Stresses on Squeeze-Film Characteristics of Porous Annular Plates REFERENCE [] Hai Wu. The Squeeze film between Rotating Porous Annular Disks. Wear, 8, 97, pp.46-47. [] Naduvinamani, N. B. Hiremath, P.S. and Gurubasavaraj, G. Squeeze film lubrication of a short porous journal bearing with couple stress fluids. Tribology international,34(),, pp. 739-747. [3] Naduvinamani, N.B. Hiremath, P.S. and Gurubasavaraj, G. Effect of surface roughness on the Static Characteristics of rotor bearings with couple stress fluids. Computers and Structures, 8,, pp. 45-458. [4] Rajesh, C. Shaha Ramesh and Katariab, C. The squeeze film characteristic between a sphere and a flat porous plate using ferrofluid. Applied Mathematical modeling, 4(3), 6, pp. 473-484. [5] Jaw Ren Lin, Rong-Fang Lu, Ming-Chung Lin, Pin-Yu Wang. Squeeze film characteristics of parallel circular disks lubricated by ferrofluids with non-newtonian couple stresses, Tribology International, 6, 3, pp 56-6. [6] Murti, P. R. K. Squeeze films in full porous metal bearings, Wear, 3, 973, pp. 57-65. [7] Bujurke, N. M. and Patil, H. P. An analysis of squeeze film action in porous layered bearings, Wear, 45, 99, pp. 385-397. [8] Ayyappa, G. H. Naduvinamani, N. B. Siddangouda, A. Biradar, S. N. Effects of viscosity variation and surface roughness on the couple stress squeeze film characteristics of short journal bearings. Tribology in Industry, 37, 5, pp. 7-7. [9] Sujatha, E. Sundarammal Kesavan. Effects of viscosity variation in porous parallel Rectangular plates lubricated with couple stress fluids, Global Journal of pure and Applied mathematics,, 6, pp. 973-768. [] Jaw Ren Lin, Lin Ming Chu, Wang-Long Li, Rong-Fang Lu. Combined effects of piezo viscous dependency and non-newtonian couple stresses is wide parallel plates squeeze film characteristics, Tribology International, 44(),, pp. 598-6. [] Martin, Reho, Vit Prusa. Squeeze flow of a peizo viscous fluid, Applied Mathematics and Computation, 74, 6, pp. 44-49. http://www.iaeme.com/ IJMET/index.asp 9 editor@iaeme.com
Noor jahan and Hanumagowda, B. N and Sreekala C.K [] Naduvinamani, N. B. Syeda thasneem fathima, Salma Jamal.Effect of roughness on hydro magnetic squeeze film between porous rectangular plates, Tribology International, 43, pp. 45-5. [3] Naduvinamani, N. B. Syeda Tasneem fathima, Hiremath P. S. Effect of surface roughness on characteristics of couple stresses film between antisotropic porous rectangular plates, Fluid Dynamics Research, 3, 3, pp.7-3. [4] Christensen H. Stochastic models for hydrodynamic lubrication of rough surfaces. Proceedings of the Institution of Mechanical Engineers. 84, 969; pp. 3 6. http://www.iaeme.com/ IJMET/index.asp 9 editor@iaeme.com