Inertia Effects in an Externally Pressurized Thrust Bearing Using Herschel - Bulkley Lubricants

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1 Ameica Joual of Computatioal ad Applied Matematics 0, (3): DOI: 0.593/j.ajcam Ietia Effects i a Exteally Pessuized Tust Beaig Usig Hescel - Bulkley Lubicats I. Jayakaa Amalaj,, S. Naasimma, A. Kadasamy Depatmet of Matematics, SSN College of Egieeig, Kalavakkam, Ceai, 6030, Idia Depatmet of Matematical ad Computatioal Scieces, Natioal Istitute of Tecology, Kaataka, Suatkal, Magaloe, 57505, Idia Abstact Te combied effects of fluid ietia ad viscous foces of a Hescel-Bulkley lubicat i a exteally pessuized tust beaig wit cicula geomety ave bee aalyzed teoetically. Altoug te eseaces of te past, laid out a foudatio fo te ydodyamic lubicatio, mode eseaces ited to use o-newtoia fluids caacteized by a yield-value, suc as Bigam, Casso ad Hescel-Bulkley fluids as lubicats. Moe ove, Tibologists empasize a fact tat i ode to aalyze te pefomace of te beaigs adequately, it is ecessay to coside te combied effects of fluid ietia ad viscous foces of o-newtoia lubicats. Teefoe, i tis eseac aticle, te combied effects of fluid ietia ad viscous foces ave bee ivestigated teoetically i a exteally pessuized tust beaig wit cicula geomety usig Hescel-Bulkley fluid as lubicat. Te sape ad extet of te coe, alog te adius, ave bee detemied umeically fo vaious values of te Hescel-Bulkley umbe ad te powe-law idex. Usig te appopiate bouday coditios, te velocity distibutios i te flow ad te coe egios ave bee obtaied. By cosideig te equilibium of a elemet of te coe i te fluid, te modified pessue gadiet as bee evaluated ad teeby te film pessue ad te load capacity of te beaig ave bee obtaied umeically fo diffeet values of Reyolds umbe, Hescel-Bulkley umbe ad powe-law idex. Te effects of te ietia foces ad te o-newtoia caacteistics of te lubicat, o te beaig pefomaces ave also bee discussed. Keywods No-Newtoia fluids, Hescel-Bulkley fluids, Exteally Pessuized Tust Beaig, Ietia effects, yield stess. Itoductio Te ecet ted i Mecaical idusty sows tat it desigs macies wit ig opeatig speeds ad eavy load cayig capacity. Teefoe, lubicatio of mode macies as bee a geat callege ad a emegig aea of iteest fo te eseaces aoud te wold. I ecet times, expeimetal eseaces ave sow clea evidece tat te use of lubicats wit vaiable viscosity ca impove te lubicat popeties elative to tat of lubicats wit costat viscosity. Teefoe, te iteest as bee iceasig to use te lubicats wit vaiable viscosity called o-newtoia lubicats, i paticula, caacteized by a yield-value kow as visco-plastic lubicats. Hece, visco-plastic lubicats ae gaiig impotace i lubicatio aalysis. Some of te impotat visco-plastic fluid models ae Bigam Plastics, Casso fluids ad Hescel-Bulkley fluids. Gos ad Majumda[7] ivestigated te effects of ecess Coespodig auto: jayakaaa@ss.edu.i (I. Jayakaa Amalaj) Publised olie at ttp://joual.sapub.og/ajcam Copyigt 0 Scietific & Academic Publisig. All Rigts Reseved volume compessibility o te dyamic caacteistics of a exteally pessuized beaigs usig Newtoia lubicat. Hsaegawa[8] as studied teoetically, te squeeze film flow of a icompessible viscous Newtoia fluid betwee a cuved disc ad a plae wall at est. Elsakawy et al.[5] itoduced a matematical model ad obtaied umeical solutios fo a exteally pessuized cicula poous beaig usig Newtoia fluid as lubicat. Seda Bais[7] applied te mixtue teoy of two cemically iet icompessible Newtoia fluids to some simple usteady flows i te aula egio betwee two ifiitely log coaxial cylides. I all tese studies, autos ave cosideed Newtoia fluids as lubicats. Recetly, te eseaces ited to use o-newtoia fluids caacteized by a yield-value called visco-plastic fluids as lubicats. Te advet of suc eological lubicats as made te teoy of lubicatio moe efficiet. Bigam, Casso ad Hescel-Bulkley fluids ae some of te visco-plastic fluids used as lubicats i te idusties. Te effects of Bigam lubicat, o slide, tust, Joual ad squeeze film beaigs ave bee aalyzed by may eseaces. Covey ad Stamoe[4] obtaied teoetical ad expeimetal esults fo te flow of Bigam fluid i a pa-

2 80 I. Jayakaa Amalaj et al.: Ietia Effects I A Exteally Pessuized Tust Beaig Usig Hescel - Bulkley Lubicats allel-plate plasto-mete ad discussed te pesece ad fomatios of yield suface. Papaastasiou[5] aalyzed te steady, two-dimesioal flow of Bigam fluid by cosideig modified costitutive equatio. Kadasamy[] aalyzed teoetically ad obtaied umeical solutios fo te beaig pefomaces of a exteally pessuized tust beaig lubicated wit o-newtoia fluids. Getzos et al.[6] developed 3-dimesioal CFD model to study te beavio ad pefomace caacteistics of a Joual beaigs lubicated wit Bigam fluid. Jaw-Re Li[], o te basis of te mico-cotiuum teoy, as ivestigated te squeeze film caacteistics betwee a spee ad a flat plate based o couple stess fluid model. Te Hescel-Bulkley fluid model epesets bot sea tiig ad sea tickeig mateials. It is te combiatio of bot Bigam plastic ad powe-law fluids ad is also kow as yield-powe law fluids. Examples of suc mateials ae geases, colloidal suspesios ad stac pastes. Te blood also beaves like Hescel-Bulkley fluids, wile flowig toug aow tubes. Catuai ad Naasimma[3] ave teoetically ivestigated te flow of Casso ad Hescel-Bulkley fluids i a coe-plate viscomete. Adams et al.[] ave implemeted fiite elemet aalysis of a squeeze flow fo a mateial tat exibits elasto- visco-plasticity ad te umeical esults ave bee veified wit expeimetal measuemets. Meeta[4] as caied out bot teoetical ad expeimetal ivestigatio of te foce equied to squeeze a Hescel-Bulkley mateial. Zag et al.[] ave umeically aalyzed te beavio of Hescel-Bulkley fluid i betwee two ellipsoid olles. Te effects of fluid ietia ave bee aalyzed teoetically by Viswaat ad Kadasamy[0] i a cicula squeeze film beaig wit costat squeeze motio usig Hescel-Bulkley lubicat. It as bee sow tat te fluid ietia foces ideed eace te pefomaces of te beaig. Jayakaa Amalaj et.al[3] studied teoetically te pefomace of a exteally pessuized tust beaig by meas of siusoidal ijectio of te Hescel-Bulkley lubicat. Ticy[8] studied te ietia effects i ydodyamic lubicatio by a lieaizatio metod ad applied it to te squeeze film dampe beaig. Ietia effects o te opeatig caacteistics of Joual beaig ave bee aalyzed by You ad Lu[]. Bata ad Kadasamy[] ave aalyzed te ietia effects i squeeze film beaig usig Bigam fluid as lubicat. J.S. Roy et.al[4] ivestigated te effects of ietia foces i a exteally pessuized tust beaig wit covegig ad divegig fluid film usig visco-elastic lubicat. Usa ad Vimala[9] ave discussed about ietia effects i cicula squeeze films cotaiig a cetal ai bubble. Te flow of a squeeze film cotaiig cetal ai bubble ude siusoidal squeeze motio as bee ivestigated by Hasimoto[9] usig te metod of aveagig ietia tems. I te peset wok, te effects of ietia i a exteally pessuized tust beaig lubicated wit Hescel-Bulkley fluid ave bee aalyzed. Duig te opeatios of te beaigs, te maximum viscous seaig stesses aise i te egio betwee te plates. Teefoe, tee may be a egio i te fluid film wee te seaig stesses do ot exceed te yield value of te lubicat ad tee by a coe wit zeo velocity gadiets is fomed. Te flow occus i te egio wee te sea stess exceeds te yield value. Te sape ad extet of te coe alog te adius fo vaious values of te Hescel-Bulkley umbe ad te powe-law idex ave bee umeically detemied. Te flow is cofied to te egio betwee te coe ad te cicula plates of te beaig. Numeical solutios ave bee obtaied fo te beaig pefomaces suc as pessue distibutio ad load capacity fo diffeet values of Hescel-Bulkley umbe, powe-law idex ad Reyolds umbe.. Fomulatio of te Poblem Te Geomety of te poblem is as sow i Figue. Figue. Geomety of a Exteally Pessuized Tust Beaig Coside a isotemal, icompessible steady flow of a time idepedet Hescel-Bulkley fluid ijected betwee two cicula plates, of te beaig, sepaated by a distace,. Let R be te adius of te film ilet, ad R be te adius of te film outlet. p, epesets te pessue of te film, p a epesets te atmospeic pessue ad ρ te desity of te fluid. Te cylidical pola coodiates (,θ, z) wit axial symmety ave bee cosideed. Te oigi is fixed at te cete of te film cleaace,, measuig te distace alog te adial diectio ad z, alog te axis omal to te beaig. Let v ad v z be te velocity compoets alog ad z diectios espectively. Te tee dimesioal costitutive equatio of te Hescel-Bulkley fluids is give by eij = 0 if τ < η e ij τij= ( ηε η) if τ η () ε Wee τ ij ae te deviatoy stess compoets, η, η ad ae costats amely te yield stess, cosistecy idex ad powe-law idex espectively. Hee v v i j eij= epesets ate of defomatio compoets xj x i

3 Ameica Joual of Computatioal ad Applied Matematics 0, (3): ad ε = eijeij is secod stai ivaiat. Fute, fo all pactical puposes te oe -dimesioal fom of equatio () is give by τ = η η γ () z wee, γ z, epesets te sea ate. I tose egios of te film, wee te sea stess is less ta te yield value, tee will be a coe fomatio, wic will move wit costat velocity, v. Let te boudaies of te coe be Figue.. z = δ ( ) ad z δ ( ) c z = wic is sow i Figue. Sape of te Coe i a exteally pessuized tust beaig Applyig te basic assumptios of lubicatio teoy fo ti films, te equatios goveig te flow of a Hescel-Bulkley fluid i a exteally pessuized tust beaig icludig ietia foces will be v v p z τ z ρ v vz = (3) z z p = 0 (4) z ( v) vz = 0 (5) z c τ v z = η η z (6) Te equatios (3), (4) ad (6) togete wit te cotiuity equatio (5) ae to be solved ude te followig bouday coditios: v = 0 at z = ± (7) v = v, a costat, at z = ± δ ( ) (8) v v is cotiuous ad = 0 at τz = η z (9) p = p a at = R (0) 3. Solutio of te Poblem Aveagig te ietia tems i te mometum equatio (3) by assumig it to be a costat ove te film tickess, leads to ρ v vz dp τ z v vz dz = z () d z Usig te cotiuity equatio (5) ad bouday coditios (7) ad (8) i (), it is educed to v dz 0 0 ρ dp z v dz = τ () d z Fo a exteally pessuized tust beaig, te modified pessue gadiet is defied as ρ dp f vdz vdz (3) d 0 0 Usig (3) i (), it becomes τ z = f (4) z Itegatig (4), τ z = fz C (5) Substitutig τ z fom equatio (5) ito equatio (6) ad usig te bouday te coditios (7) ad (8), te velocity distibutio i te flow egio will be f v = δ ( δ ) z, η δ z (6) ad te coe velocity will be f vc = ( δ ), η δ 0 z (7) Cosideig te equilibium of a elemet of te coe i te fluid, equatio (4) becomes η f = ( ) (8) δ Wee δ ( ) epesets te tickess of te coe. Te equatio of cosevatio of mass wic depeds o te cofiguatio of te beaig is give by Q = 4 π v dz 0, (9) Usig te velocity expessios (6) ad (7) i (9), it is foud tat ( )( ) η Q f = (0) π δ δ ( ) ( )

4 8 I. Jayakaa Amalaj et al.: Ietia Effects I A Exteally Pessuized Tust Beaig Usig Hescel - Bulkley Lubicats Elimiatig f fom equatios (8) ad (0), a algebaic equatio fo detemiig te tickess of te coe as bee obtaied as, / Q / / δ π η ( δ) ( δ ) ( )( ) η = 0 () Fute, substitutig te expessios (6), (7) ad (0) i (), ( )( ) ( ) dp η Q ρq = 3 d δ π δ 4π ( k k ) () ( ) Wee 3 ( ) ( 4 3) δ ( 6 5) δ ( ) k = 3 ( 3 ) ( δ ) dδ ( ) ( 4 3) δ ( ) d k = 3 ( 3 )( δ ) By itoducig te followig o-dimesioal quatities, pπ R pπ R =, δ = δ ( ), p =, p a = R Q η Q η te o-dimesioal fom of equatio () ca be expessed as ( δ ) ( δ ) ( δ ) ( ) ( ) = 0 (4) / N / η / η π R Wee N = is called te yield umbe (o) Q Hescel-Bulkley umbe. Te coe tickess ca be detemied fom te above o-liea algebaic equatio usig ay iteative tecique. wic is positive ad less ta uity, detemies te yield suface, betwee te plates. Te coe tickess is detemied fo vaious values of, ad N. Te pessue gadiet expessio i o-dimesioal fom is obtaied as dp ( )( ) = Re k k d ( δ ) δ (5) Wee 3 ( ) ( 4 3) δ ( 6 5) δ ( ) k = δ Te oot of equatio (4), δ ( N, ) k = ( ) ( ) ( ) ( 4 3 ) δ ( ) 43 ( ) ( δ ) 3 3 dδ d Te expessio fo te pessue distibutio ca be obtaied by itegatig (5) ad usig te bouday coditio (7) as ( )( ) P Pa = Re k k d (6) ( δ ) δ ad Re = ρ Q is te Reyolds umbe. 3 π ηr Te load capacity fo a exteally pessuized tust beaig is, W = p p d (7) ( a ) R wee R R =, (i.e) R is te atio of te ilet to te outlet R of te beaig. Te above itegal is evaluated umeically fo vaious values of powe-law idex, Reyolds umbe ad Hescel-Bulkley umbe. 4. Results ad Discussio Te tickess of te coe alog te adial diectio fo vaious values of powe-law idex( ) ad Hescel-Bulkley umbe ( N ) ave bee computed umeically ad ae sow i te Figues Figue 3. Coe tickess vaiatio alog te adius fo = 0.7 Figue 4. Coe tickess vaiatio alog te adius fo =.0

5 Ameica Joual of Computatioal ad Applied Matematics 0, (3): Figue 8. Ietia effects o Pessue distibutio fo =.0 Figue 5. Coe tickess vaiatio alog te adius fo =.3 Te plug coe occus symmetically i te middle of te egio betwee te cicula plates. Te tickess of te coe alog te adial diectio iceases fom cete to te peipey. It as also bee obseved tat fo a give powe-law idex, te tickess of te coe iceases wit te icease i Hescel-Bulkley umbe. Moeove, tee is a icease i te coe tickess wit icease i powe-law idex fo a paticula Hescel-Bulkley umbe. Te velocity pofiles fo vaious Hescel-Bulkley umbe umbes alog te axial diectio at diffeet adial positios ave bee obtaied ad ae sow i Figue.6. Figue 6. Velocity Pofile fo a paticula powe-law idex Figue 9. Ietia effects o Pessue distibutio fo =.3 Te pesece ad tickess of te coe is vey muc eflected i te velocity pofile. Fute, it as bee foud tat i te limitig case, as N teds to zeo, te velocity pofile becomes paabolic wic epesets Newtoia fluid. Te distibutio of film pessue i te adial diectio fo vaious values of te Reyolds umbe ( Re ), Hescel-Bulkley umbe ad powe-law idex, ad ae sow i te Figues Te pessue is foud to be ig at te cete ad deceases towads te peipey of te beaig alog te adial diectio. It as bee obseved tat tee is a sigificat icease i te pessue fo te fluids wit ig Hescel-Bulkley umbe fo a specific powe-law idex. Fute, tee is a cosideable icease i te pessue distibutio due to te icease i powe-law idex of te fluid. Te icease i pessue is also obseved as te Reyolds umbe iceases. Howeve, te amout of icease i pessue due to Reyolds umbe is magial. Moeove, te effects of ietia o pessue distibutio ae foud to be appeciable ea te cete of te beaig. Te esults of load-cayig capacity fo diffeet values of powe-law idex, Hescel-Bulkley umbe ad Reyolds umbe ae tabulated i te Tables 3. Table. Load capacity values fo N ad Re fo = 0.7 Figue 7. Ietia effects o Pessue distibutio fo = 0.7 N/Re

6 84 I. Jayakaa Amalaj et al.: Ietia Effects I A Exteally Pessuized Tust Beaig Usig Hescel - Bulkley Lubicats Table. Load capacity values fo N ad Re fo =.0 N/Re Table 3. Load capacity values fo N ad Re fo =.3 N/Re It as bee obseved tat te load cayig capacity of te beaig iceases wit te icease i Hescel-Bulkley umbe fo a specific powe-law idex. Fo a paticula Hescel-Bulkley umbe, te load cayig capacity iceases sigificatly wit te icease i powe-law idex. Howeve, te effect of fluid ietia o te load cayig capacity is foud to be magial. Te pecetage of icease i load cayig capacity fo diffeet values of powe-law idex, Hescel-Bulkley umbe ad Reyolds umbe ae give i te Tables 4 6. Table 4. Pecetage vaiatio i Load capacity fo = 0.7 N/Re Table 5. Pecetage vaiatio i Load capacity fo =.0 N/Re Table 6. Pecetage vaiatio i Load capacity fo =.3 N/Re It as bee obseved tat te pecetage of icease i load cayig capacity due to ietia effect is sigificat oly fo tose fluids wit low Hescel-Bulkley umbe wee as tis pecetage of icease deceases wit te icease i Hescel-Bulkley umbe fo ay powe-law idex ad Reyolds umbe. Moeove, te pecetage of icease i te load cayig capacity iceases sigificatly as te Reyolds umbe iceases. Te esults coespodig to Re = 0 wic exibits te ietia less effects o te pessue distibutio ad te load cayig capacity of te beaig ae foud to be i good ageemet wit te esults of Kadasamy et.al[3]. Similaly, te esults coespodig to =, ae i ageemet wit te esults obtaied by Jayakaa Amalaj et.al[0]. 5. Coclussios Te above teoetical ivestigatio eveals tat te icease i Hescel-Bulkley umbe ad powe-law idex sigificatly eace te beaig pefomaces suc as pessue distibutio ad load cayig capacity. Howeve, te effect of fluid ietia o te beaig pefomaces is foud to be magial tat itself ea te cete of te beaig. ACKNOWLEDGEMENTS Te autos () ad () wis to tak te maagemet of SS college of Egieeig fo povidig te ecessay facilities, suppot ad ecouagemet to cay out tis eseac wok. REFERENCES [] M. Adams, B. Edmodso, D.G. Caugey, ad R. Yaya., A expeimetal ad teoetical study of te squeeze film defomatio ad flow of a elasto-plastic fluid., J. o-newtoia Fluid Mecaics., Vol. 5, No., pp [] R.L. Bata ad A. Kadasamy,. Ietia effects Redyamic lubicatio of a squeeze film beaig. Wea. Vol. 3, pp. 73 8, 989. [3] P. Catuai ad S. Naasimma.,. Teoy fo Casso ad Hescel-Bulkley fluids i coe-plate viscomete, Bioeology, Vol. 5, pp. 99, 988. [4] G.H. Covey ad B.R. Stamoe., Use of paallel-plate plasto-mete fo te caacteizatio of viscous fluids study wit a yield stess, J. o-newtoia Fluid Mecaics. Vol. 8, pp , 98. [5] A.A. Elsakawy, S.Z. Kassab ad M.M. Nassa, Lubicatio aalysis of exteally pessuized cicula poous beaig, Applied Matematical Modellig Vol. 0, pp , 996. [6] K.P. Getzos, P.G. Nikolakopoulos ad C.A. Papadopoulos, CFD aalysis of joual beaig ydodyamic lubicatio by Bigam lubicat. Tibology Iteatioal. Vol. 4, pp , 008. [7] M.K. Gos ad B.C. Majumda., Dyamic stiffess ad dampig caacteistics compesated ydostatic tust beaigs. ASME, J. Lubicatio Tecology, Vol. 04, pp , 98. [8] E. Hasegawa., Squeeze film due to a otatig cuved disk

7 Ameica Joual of Computatioal ad Applied Matematics 0, (3): appoacig a fixed plae ude te actio of a foce ad momet. Wea. Vol. 6, No., pp , 987. [9] H. Hasimoto., Squeeze film caacteistics betwee paallel cicula plates cotaiig a sigle cetal ai bubble i te ietial flow egime. J. Tibology, Vol. 7 No. 3, pp , 995,. [0] I. Jayakaa Amalaj, A. Kadasamy, ad S. Naasimma., Ietia effects i Reodyamic lubicatio of a exteally pessuized tust beaig usig Bigam fluids, Poc. It. co. ACFD 8 Ja. (00), Uiv. of Sci. ad Tec., Hog kog. [] Jaw-Re Li., Squeeze film caacteistics betwee a spee ad a flat plate couple stess fluid model. Computes & Stuctues., Vol. 75, No., pp , 000. [] A. Kadasamy., Lubicatio of a exteally pessuized tust beaig wit o-newtoia lubicats. Poc. It. Cof. o Advaces i Mecaical Egieeig, Bagaloe, Idia, pp , 995. [3] A. Kadasamy, S. Naasimma ad I. Jayakaa Amalaj., Reodyamic lubicatio of a exteally pessuized tust beaig wit Hescel-Bulkkley fluids, Poc. It. Cof. o idustial Tibology, IISc, Bagaloe, 006. [4] G.H. Meete, Flow of soft solids squeezed betwee plaa ad speical suface, Reol. Acta., Vol. 44, pp , 005. [5] T.C. Papaastasiou., Flows of Mateials wit Yield. J. Reology., Vol. 3, No. 5, pp , 987. [6] J. Sia Roy, S. Pady ad L.K. Bopa,, Iteia effects i exteally pessuized tust beaig wit covegig ad divegig film usig visco-elastic lubicat. Acta Mecaica, Vol. 96, pp. -, 993. [7] Seda Bais., Usteady flows of a biay mixtue of icompessible Newtoia fluids i a aulus. It. J. Egieeig Sciece. Vol. 43, pp , 005. [8] J.A. Ticy., Te effect of fluid ietia i ydodyamic lubicatio Studied by a lieaizatio metod ad Two useful applicatios. It. J Egieeig Sciece. Vol. 4, No. 0, pp , 986. [9] R. Usa ad P. Vimala., Ietia effects i a cicula squeeze films cotaiig a cetal ai bubble. Fluid Dyamic Reseac, Vol. 6, No. 3, pp , 000 [0] K.P.Viswaat ad A.Kadasamy., Ietia Effects i Cicula Squeeze Film Beaig Usig Hescel-Bulkley Lubicat., Applied Matematical Modelig., Vol.34, pp. 9 7, 00. [] H.I. You ad S.S. Lu., Te effects of fluid ietia o te opeatig Caacteistics of a joual beaig. J. Tibology. Vol. 0, No. 3, pp , 988. [] J. Zag, K.E. Kayat ad A.P. Nooa., Tee dimesioal lubicatio flow of a Hescel-Bulkley fluid. It. J. Numeical Metods i Fluids. Vol. 50, pp , 005. [3] I. Jayakaa Amalaj, S. Naasimma. ad A. Kadasamy., Reodyamic Lubicatio of a Exteally pessuized tust beaig usig Hescel Bulkley fluid wit siusoidal ijectio., J. Applied Fluid Mecaics., Vol.6, No., 03 (To be publised).