Effects of piezo-viscous dependency on squeeze film between circular plates: Couple Stress fluid model

Transcription

1 Turkish Jurnal f Science & Technlgy Vlume 9(1), , 014 Effects f piez-viscus dependency n squeeze film between circular plates: Cuple Stress fluid mdel Abstract U. P. SINGH Ansal Technical Campus, Faculty f Applied Science, Department f Mathematics, Lucknw; India; * jurnals4phd@gmail.cm (Received: ; Accepted: ) In high pressure fluid flws applicatins such as fluid film lubricatin, micrfluidics and gephysics, the piezviscus effect i.e. viscsity-pressure dependence plays an imprtant rle. In the present theretical investigatin, the cmbined effects f piez-viscus dependency and nn-newtnian cuple stresses n the perfrmance f circular plate squeeze film bearings have been investigated using Stkes Micr-cntinuum thery f cuple stress fluids tgether with the expnential variatin f viscsity with pressure. Analytic slutin fr film pressure is btained using small perturbatin analysis. The numerical results fr pressure and lad capacity with different values f viscsity-pressure parameter are calculated and cmpared with is-viscus cuple stress and Newtnian lubricants. Due t piez-viscus effect, enhanced pressure, increased lad capacity and lnger respnse time is bserved in the analysis. Key wrds: Cuple stress Fluids, Lad-carrying capacity, Piez-viscus effect, Squeeze-film lubricatin, Viscsity-pressure 1. Intrductin The squeeze film lubricatin plays very imprtant rle in varius applicatins f engineering and technlgy. Ball bearings, matching gears, machine tls, rlling elements and autmtive engines are sme cmmn examples f squeeze film lubricatin. The squeeze film phenmenn is als bserved during apprach f faces f disc clutches under lubricated cnditin. The mechanical actin (squeezing, shearing etc.) changes the lubricants temperature as well as viscsity and density which accunt fr the variatin f bearings perfrmance characteristics. Many researchers such as Dwsn [1], Wada and Hayashi [] and Kapur [3] emphasized the variatin f viscsity and density with temperature and pressure and reprted a significant variatin in the pressure, lad capacity etc. In high pressure applicatins, the viscsity f lubricants depends much n pressure than the temperature [4]. In such mechanism, the variatin f viscsity is als mre significantly ver the density variatin [4, 5]. Denn [6] emphasized that under a pressure f abut 5 MPa, the dependence f viscsity n pressure becme imprtant even if the flw is incmpressible. Bair et al. [7] shwn that the viscsity f bis(phenxyphenxy)benzene varies frm Pa.s t 7 Pa.s under pressure variatin MPa, and the viscsity f dipentaerythritl hexaisstearate varies frm Pa.s t 66. Pa.s under pressure variatin MPa under isthermal cnditin. Therefre, it is reasnable t analyze the bearing perfrmance with a pressure dependent viscsity [8] cnsidering the isthermal, incmpressible flw f lubricants f the frm p e (1) where is the viscsity, P is the pressure, is the viscsity at atmspheric pressure, and is the pressure-viscsity cefficient. Kttke [9] reprted the range f between 10 t 70 MPa 1 fr lubricants. Venner and Lubrecht [10] reprted that may range between 10 8 t 10 8 Pa 1 fr mineral ils. The investigatin f squeeze film characteristics between rectangular, prus rectangular, curved circular plates etc. has been analyzed frm time t time by varius researchers such as Burbidge and Clin [11] Sanni [1], Abell and Ames [13], Gupta and Vra [14], Gupta and Gupta [15], Murti [16] and Wu [17] using classical Newtnian mdel. After Stkes [18] prpsed micr-cntinuum thery fr cuple stresses in fluids accunting the particle size effects f micr- structures in

2 Effects f piez-viscus dependency n squeeze film between circular plates : Cuple Stress fluid mdel lubricants, many researchers investigated the effect f cuple stresses n the squeeze film characteristics f varius systems. Manivasakan and Sumathi [19] studied the effect f cuple stress n squeezing films between curved circular plates. Nadumani et al. [0, 1] studied the cuple stress effect n squeezing actin between stepped circular plates and sphere-plate system with rugh surface. Lin et al [,3] analyzed the squeeze film characteristics between tw different spheres, wide parallelplate squeeze film, and the cylinder-plate squeeze film. Alkuh and Der-Fa [4] studied the nn-newtnian effects n Rayleigh stepbearings. Ishizawa [5] analyzed the unsteady flw between tw parallel disks with varying gap. As discussed earlier, the viscsity-pressure dependence is imprtant while studying the high pressure phenmena in lubricatin. In recent years, researchers als fcused their attentin t accunt the viscsity-pressure dependence t analyze the lubricatin phenmenn. Guld [6] analyzed the squeezing characteristic in high pressure spherical system and reprted an enhanced characteristics due variatin f viscsity with pressure. Yadav and Kapur [3] reprted an enhanced pressure and lad capacity f hydrstatic thrust bearings cnsidering the viscsity variatin with pressure and temperature. Kalgiru [7] analyzed incmpressible Piseuille flws f Newtnian liquids with a pressure-dependent viscsity. Lu and Lin [7,8] investigated the viscsitypressure dependence n sphere plate squeezing system and wide parallel rectangular pelate with nn-newtnian cuple stresses lubricant and reprted an enhanced film pressure and lad capacity due t piez-viscus effect. Hwever, the piez-viscus effect n the squeezing characteristics between parallel circular plates with nn-newtnian cuple stress effect was nt cnsidered by any f the authrs, which is cnsidered in the present theretical analysis.. Analysis A systematic diagram f squeeze film lubricatin between circular plates appraching each ther with a nrmal velcity dh is shwn in Fig. 1. The lubricant in the system is cnsidered as nn-newtnian incmpressible cuple stress fluid [18]. The bdy frces and bdy cuples are assumed t be absent in the analysis. Figure 1. Systematic diagram f circular plate squeeze film bearing. Under the assumptins f thin film lubricatins [1], the cnstitutive equatins in cylinder plar crdinates gverning the mtin f steady laminar flw f a cuple stress fluid in the film regin are - 1 w ( ru) 0 r r z u u p z z r p 0 z () (3) (4) which are slved under the fllwing bundary cnditins : u u z u u z 0 0 and and w 0 at z 0 (5) h w t at z h (6) where u and w are the velcity cmpnents in x and z directins, h is the film thickness between the bearings plates, is viscsity given by equatin (1.1) and is new material cnstant respnsible fr cuple stresses in the fluids. Taking the dimensinless quantities as 98

3 U. P. Singh 3 p h r z p, r, z, dh R h R u R u h w,, w dh dh R ( dh ) h, h, 3 h h and 1 h the equatins (1-4) takes the dimensinless frm e p (7) 1 w ( ru) 0 r r z u u p z z r p 0 z and the related bundary cnditins (5-6) as u u z u u z 0 0 (8) (9) (10) and w 0 at z 0 (11) and w 1 at z h (1) Slving equatins (9) under the relevant bundary cnditins (11-1), the radial velcity prfile is btained as h Csh z 1 p p u z z h 1 r r h Csh (13) Integrating equatin (8) fr w with the relevant bundary cnditins (11-1) using equatins (7) and (13), the mdified Reynlds equatin is btained as d dp r f h,,, p 1r (14) dr dr where 3 p p f ( h,,, p) h e 1h e 5 p 1 3 h p 4 e tanh e (15) Due t the nn-linear nature f the mdified Reynlds equatin (14), the classical perturbatin technique is adpted t find the 99 clse frm slutin fr the pressure distributin. Since the piez-viscus parameter is small ( 1) [10], the pressure p can be written as p p p p... (16) 1 Substituting equatin (16) in the mdified Reynlds equatin (14) and leaving the terms with the secnd and higher pwers f, the fllwing simplified equatins are btained as- d dp r 1r (17) dr dr f d dp1 f1 d dp r f r p (18) dr dr dr dr Slving the equatins (17) and (18) under the bundary cnditins dp 0 at r 0 (19) dr p 0 at r 1 (0) the fllwing perturbed slutin are btained 3 p (1 r ) (1) p 1 f 9 f1 3 (1 r ) f () where 3 3 f h 1h 4 tanh( h ) (3) 3 3 f1 h 30h 60 tanh( h ) (4) 6 h tanh ( h ) The lad carrying capacity f the bearing can be calculated as fllws R W r p dr (5) 0 which takes the dimensinless frm h3 1 W W r p dr R4 dh (6) 0 The respnse time-height relatin can be calculated by integrating the equatin (6) under the cnditin h h at t 0 (i.e., h=1 at t=0) as fllws wh 1 1 t 0 R 4 t rpdr dh h (7) Since, the integrand f t is a nnlinear functin f h, Numerical integratin has been perfrmed t calculate time-height relatin using Mathematica 7.0.

4 Effects f piez-viscus dependency n squeeze film between circular plates : Cuple Stress fluid mdel 3. Results and Discussin In the present paper, the cmbined influence f cuple stresses and the piez-viscus dependency n the squeeze film bearing characteristics have been predicted n the basis f Stkes cuple stress fluid thery [18] tgether with the expnential variatin f viscsity with pressure [8] aviding the inertia and cavitatin effects. The piez-viscus effect is analyzed with a viscsity-pressure parameter n ne hand and the effects f cuple stresses are analyzed using a dimensinless parameter 1 R n the ther, where has the dimensin f length and it can be identified as the functin f mlecular length f plar additives in a nnplar lubricant. In figure t figure 7, the results f pressure, lad capacity and respnse time are cmpared fr piez-viscus and is-viscus lubricants. The ranges f parameters are taken as : piez-viscus parameter =0.0 and cuple stress parameter = 0, 0.05, 0.1, Figure shws the variatin f dimensinless pressure p with respect t the dimensinless radius r fr film thickness h On cmparisn with the is-viscus Newtnian case ( 0), the pressure with isviscus cuple stress is higher and it increases with increase f cuple stress parameter fr each value f the radius. It establishes the validity f present analysis fr is-viscus cuple stress lubricant. Further, it is bserved that fr each value f cuple stress parameter, the pressure with piz-viscus analysis is significantly higher than the pressure with isviscus analysis. Hwever, the relative variatin f pressure with piez-viscus effect increases with the increase f cuple stress parameter. It prves the significance f viscsity-pressure variatin fr cuple stress lubricants. Figure. Variatin f dimensinless pressure with respect t dimensinless radius fr is-viscus and piez-viscus lubricants. Figure 3 shws the variatin f dimensinless maximum film pressure with respect t the dimensinless film thickness h. It is bserved that the maximum film pressure with cuple stress lubricants is higher than that in Newtnian case. The maximum pressure als increases with the increase f cuple stress parameter. Again, fr each value f cuple stress parameter, the maximum pressure in pizviscus case is higher than that in the is-viscus case. Figure 3. Variatin f dimensinless maximum pressure with respect t film thickness fr is-viscus and piez-viscus lubricants. 100 Figure 4 shws the variatin f dimensinless lad capacity with respect t the dimensinless film thickness h. It is bserved that the lad capacity btained with the is-

5 U. P. Singh viscus cuple stress lubricants is higher than that f is-viscus Newtnian lubricants. The lad capacity als increase with the increase f cuple stress parameter. Further, fr each value f cuple stress parameter, the piez-viscus effect increases the lad capacity significantly. A sequential change frm is-viscus Newtnian t is-viscus cuple stress and is-viscus t piez-viscus cuple stress increases the lad capacity systematically. Figure 5 shws the variatin f dimensinless lad capacity with respect t the piez-viscus parameter ( ) fr dimensinless film thickness h =0.5. In bth the cases f Newtnian and Cuple stress lubricants, the lad capacity increases with the increase f piezviscus parameter. Hwever, the relative change in the lad capacity is bserved t increase with the increase f cuple stress parameter. The values f the lad capacity fr is-viscus case are bserved t be lwest. Again, the lad capacity fr cuple stress is bserved t be higher than the Newtnian and increases with the increase f cuple stress. fluids and fr the same value f cuple stress parameter with piez-viscus effect, a further increase in the respnse time is bserved. Again, the respnse time increases with the increase f cuple stress parameter and fr each value f cuple stress parameter, the respnse time in piez-viscus case is lnger than that in the isviscus case. Since, the higher film pressure results in a lnger respnse time, the effects f bth the cuple stresses and the piez-viscus dependency increase the respnse time. Figure 7 shws the variatin f dimensinless respnse time t t attain a particular dimensinless film thickness h =0.5 with respect t the piez-viscus parameter ( ) fr different values f cuple stress parameter. The increase in the respnse time is bserved due t the piez-viscus effect fr Newtnian as well as the Cuple stress lubricants. The effect f higher value f piez-viscus parameter prduces relatively lnger change in the respnse time fr larger values f cuple stress parameter. This behaviur is due t the higher film pressure generated by the higher value f cuple stress. Again, the effect f cuple stresses increase the respnse time significantly in cmparisn with the Newtnian. Figure 4. Variatin f dimensinless lad capacity with the dimensinless film thickness (h) fr isviscus and piez-viscus analysis. Figure 6 shws the relatin between the dimensinless film thickness ( h ) and the dimensinless respnse time ( t ) in is-viscus and piez-viscus cases fr different values f cuple stress parameter. It is clearly bserved that fr each value f film thickness, the respnse time fr is-viscus cuple stress fluids is lnger than in the is-viscus Newtnian 101 Figure 5. Variatin f dimensinless lad capacity with piez-viscus parameter fr different values f cuple stress parameter. 4. Cnclusins Based n Stkes micr-cntinuum thery f cuple stress fluids and the viscsity-pressure dependence, the analytical slutin fr pressure

6 Effects f piez-viscus dependency n squeeze film between circular plates : Cuple Stress fluid mdel distributin, lad capacity and respnse time is btained using Hmtpy Perturbatin Methd. In the present theretical analysis, fllwing results have been drawn- The effect f piez-viscus dependency as well as the cuple stresses enhances the pressure and lad carrying capacity significantly. The effects f bth the cuple stresses and piez-viscus dependency increase the respnse time. A small value f piez-viscus parameter ( 0.0 increases the pressure and lad capacity by nearly 15% and increase the respnse time apprximately 1% in cmparisn with the is-viscus case. Figure 6. Variatin f dimensinless respnse time with the dimensinless film thickness (h) fr isviscus and piez-viscus analysis. Figure 7. Variatin f dimensinless respnse time with respect t fr different values f. 5. References 1. Dwsn D., (1961). Inertia effect in hydrstatic thrust bearings, Jurnal f Basic Engineering: Transactin f ASME (Series D), 83(), Wada, S. and H. Hayashi (1971). Hydrdynamic lubricatin f jurnal bearings by Pseudplastic lubricants. Bulletin f JSME-14, 69, Yadav J. S. and V. K. Kapur (1981) On the viscsity variatin with temperature and pressure in thrust bearing. Internatinal Jurnal f Engineering Science, 19(), Rajagpal K. R., (006). On implicit cnstitutive theries fr fluids. J. Fluid Mech., 550, Renardy M. (003). Parallel shear flws f fluids with a pressure-dependent viscsity. J. Nn- Newtnian Fluid Mech., 114, Denn M. M. (008). Plymer Melt Prcessing. Cambridge University Press. Cambridge. 7. Bair C., Y. Liu and Q. J. Wang, (006). The pressure-viscsity cefficient fr Newtnian EHL film thickness with general piez-viscus respnse. Jurnal f Triblgy, Trans. ASME, 18, Kalgiru A., S. Pyiadji and G. C. Gergiu (011). Incmpressible Piseuille flws f Newtnian liquids with a pressure-dependent viscsity. J. Nn-Newtnian Fluid Mech., 166, Kttke P. A., (004). Rhelgical implicatins f tensin in liquids. Ph.D. Thesis, Gergia Institute f Technlgy. 10. Venner C. and A. A. Lubrecht (000). Multilevel Methds in Lubricatin. Elsevier. 11. Burbidge A. S. and C. Servais (004). Squeeze flws f apparently lubricated thin films. Jurnal f nn-newtnian Fluid Mechanics, 14, Sanni S. A., (1994). Squeezing flw between varying-permeability prus rectangular plates. Indian Jurnal f Pure and Applied Mathematics, 5, Abell M. L. and W. F. Ames (199). Symmetry reductin f Reynlds equatin t film lubricatin. Jurnal f Applied Mechanics, 59, Gupta J. L. and K. H. Vra (1980). Analysis f squeeze films between curved annular plates. Jurnal f Lubricatin Technlgy, 10, Gupta P. S. and A. S. Gupta (1977). Squeezing flw between parallel plates. Wear, 45(), Murti P. R. K., (1975). Squeeze films in curved circular plates. Jurnal f Lubricatin Technlgy, 97, Wu H., (197). An analysis f the squeeze films 10

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