On the modified Reynolds equation for journal bearings in a case of non-newtonian Rabinowitsch fluid model

Transcription

1 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 On te modified Renolds equation for journal bearings in a case of non-newtonian Rabinowitsc fluid model Juliana Javorova,*, and Jordanka Angelova UCTM, Dept. of Applied Mecanics, 8 Kliment Oridski Blvd., 756 Sofia, Bulgaria UCTM, Dept. of Matematics, 8 Kliment Oridski Blvd., 756 Sofia, Bulgaria Abstract. In tis paper, a teoretical analsis of drodnamic plain journal bearings wit finite lengt at taking into account te effect of non- Newtonian lubricants is presented. Based upon te Rabinowitsc fluid model (cubic stress constitutive equation) and b integrating te continuit equation across te film, te nonlinear modified D Renolds tpe equation is derived in details so tat to stud te dilatant and pseudoplastic nature of te lubricant in comparison wit Newtonian fluid. A dimensionless equation of drodnamic pressure distribution in a form appropriate for numerical modeling is also presented. Some particular cases of D applications can be recovered from te present derivation. Introduction Hdrodnamic journal bearings are considered to be a vital component of all rotating macines. Journal bearings are used widel for tousands of ears in man areas of mecanical engineering. Currentl large numbers of tem are applied successfull in macine tools, automotive and aircraft piston engines, turbo-maciner, micro-electromecanical sstems, etc. Furtermore, it is well known tat te performance of bearings is strongl influenced b te composition and reological caracteristics of te lubricant. In order to meet requirements of te modern maciner sstems, it is necessar to look for te enancement in lubricating performance of all kind of bearings and oter tpes of lubricated contacts. In tis relation, currentl, te use of Newtonian fluids blended wit various additives increases, wic is based on te effective improvement in te bearing caracteristics as compared to te lubrication wit Newtonian lubricants. Tese additives often are viscosit index improvers wic represent ig molecular weigt polmers suc as polisobutlene, polmetacrlate, etlene proplene, etc. Tese kinds of lubricants exibit pseudoplastic and dilatant beaviour of non-newtonian lubricants, in wic te ratio between te sear stress and sear rate is no longer a constant. According to te experimental work of Wada and Haasi [] te non-newtonian reological beaviour of suc kinds of lubricants wit additives can be represented b an empiric cubic stress model, also called Rabinowitsc fluid model. * Corresponding autor: jul@uctm.edu Te Autors, publised b EDP Sciences. Tis is an open access article distributed under te terms of te Creative Commons Attribution License 4. (ttp://creativecommons.org/licenses/b/4./).

2 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 Appling tis fluid model, a number of scientific papers concerning te subjects of non- Newtonian fluid film lubrication ave been publised. For instance, te teoretical stud of bearing performance wit non-newtonian lubricants using Rabinowitsc fluid model is done on journal bearings b Wada and Haasi [], Rajalingam et al. [], Bourgin and Ga [], Sarma et al. [4], Javorova et al. [5, 6], Kusare [7], Lin et al. [8]. Some contributions for squeeze film bearings are presented b Hasimoto and Wada [9], Lin [], Naduvamani [], Sigealli []. Furtermore, Lin et al. [-5] and Sing et al. [6-] used te model of Rabinowitsc fluid to investigate in details te performance of different tpes of drostatic, drodnamic and squeeze film bearing sstems. More general lubrication problems for te brid bearings can be found in te works of Walicka et al. [, ], Ratajczak et al. [] as te same autors work also on trust bearings [4]. Recentl several papers in te field of peristaltic flow in a tube wit Rabinowitsc fluid model are also publised. As a wole, according to results in te mentioned studies, te influences of non- Newtonian Rabinowitsc fluids on te lubrication performances of journal bearings, slider bearings, squeeze film bearings, drostatic bearings, and oter tpes of lubricated contacts are significantl apparent. It is sown tat te stead state and squeeze-film performances are strongl affected b non-newtonian effects based on te Rabinowitsc fluid model. Te most of above mentioned papers aimed to obtain some modified Renolds equation for suc kind of lubricants as well as to receive an analtical solution of it for more simple cases. For more complicated ones, as beaviour of finite lengt journal bearings, te autors regularl appl modified Renolds equation obtained b Wada and Haasi derived in sort in []. In te mentioned studies a detailed matematical output of modified Renolds equation is not given tat motivates autors to present ere suc one. Te current stud aims to represent consistentl and in details a matematical procedure to obtain te nonlinear modified two-dimensional Renolds tpe equation based on te Rabinowitsc fluid model. B tis wa it is possible to stud te pseudoplastic and dilatant nature of lubricants wit long cain polmer additives in comparison wit Newtonian fluids. Modeling of te drodnamic lubrication in journal bearings A radial journal bearing wit finite lengt is considered under stead state conditions. Te lubricant fluid in bearing clearance as non-newtonian properties. Te reological fluid law described b te Rabinowitsc model [,, 8, etc.] is presented wit te following cubic equation du k, () d were: - sear stress in te fluid film ; k - coefficient of pseudoplasticit (parameter responsible for te lubricants non-newtonian beaviour); - initial viscosit of te lubricant; uvw,, - velocit components in te directions of Cartesian coordinates xz,,, respectivel. In dependence of te values of te coefficient of pseudoplasticit k tere are tree different groups of lubricants: for k - pseudoplastic fluids, if k - Newtonian fluid, as for k - dilatant fluid.

3 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 Fig.. Flow curves for non-newtonian Rabinowitsc fluid. Te flow caracteristics of tese kinds of lubricants are sown in Fig.. Here is a tangent at te original point of flow curves because of wic it is called initial viscosit b analog wit [,, 7]. If values of do not var, te nonlinearit of te flow curve increases wit te value of te coefficient of pseudoplasticit k (see Fig. - on te rigt).. Bearing geometr For te geometr of te considered plane journal bearing wit 6 range are introduced te following assumptions (see Fig.): Te journal and te bearing ave a round sape and parallel axes; teir surfaces are perfectl smoot. Te bearing gap is filled wit a lubricating fluid wit a constant pressure equal to te external. Te journal rotates wit a constant angular velocit around its axis. Te bearing radius R is approximatel equal to te journal radius r R r. Te radial bearing clearance c R r is of te order 4., i.e. c, see Fig.. Terefore te clearance ratio bearing eccentricit is also ver small, so te eccentricit ratio cr.. Te er.. Moreover, for all tpes of bearings te fluid film tickness is ver small compared to te oter dimensions of te contact surfaces because of wic te ratio lr., were l is te fluid film lengt in a circumferential direction. All of te above mentioned about te bearing geometr justifies te commonl accepted potesis in te teor of drodnamic lubrication to neglect te curvature of te lubricated surfaces and, respectivel, te curvature of te lubricant film. Fig.. Hdrodnamic journal bearing.

4 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457. Boundar value problem Te constitutive equations between te sear stress and rate of sare strain for two dimensional flow of te Rabinowitsc fluid obes te following nonlinear relationsips du x kx, (.a) d dw z kz, (.b) d were x and z are sear stress components in x and z directions, respectivel. According to te tin film teor of drodnamic lubrication [5,, 8], te momentum and continuit equations in Cartesian coordinates are represented b te following differential equations: p x ; (.а) x p ; (.b) p z z ; (.c) u v w, (.d) x z were p is te drodnamic pressure. Related to te bearing configuration, te boundar conditions for te fluid velocit components are at : at :,, ; v x,, z v ; w x z w u x z u ; u xz,, u U r v xz,, v r x were: is te journal angular velocit, xz,,,, (4) ; w xz,, w, (5) is te fluid film tickness.. Matematical procedure Te integration of (.a) and (.c) wit respect to ields: x z p ; (6.a) x p, (6.b) z 4

5 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 were and are integration constants. B substitution of (6.a) and (6.b) in (.a) and (.b), respectivel, it is obtained: df d q k q, (7) were f uw,, qp x, p z,,. Here te couples mapping between tem one-to-one. Integration of (7) wit respect to, were: f u, w, f u, w leads to..,.. are ordered and f f q k q d, (8). Appling boundar conditions (4) and (5) and performing some elementar transformations te following cubic equation for is obtained: q q f q q k. (9) 4 k k On te oter and, reordering (9) in ascending order b power of a semi-cubic equation of : q leads to q f. () k k If f /see (4) for q w /, ten, and q, i.e. q. u According to [], since te value of is enoug small witin te k q q range of parameters considered, ten from (8) follows tat,. s, it is possible to find an Furtermore, b integrating (7) wit respect to antiderivative of f as f f qs ds kqs ds, () were s - variable of integration. Afterwards, based on () te velocit component u is obtained in a form: 5

6 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 q u q s ds k qs ds q k q s ds q s ds q s ds. () Neglecting in () terms multipling b second and tird powers of ields u q kq kq, () were: ; s ds ; (4) s ds 4 4. s ds 4 Te expression for w is similar to () witout te term multipl b. Terefore u kq p u q kq, q ; (5) kq x p w q kq, q z. (6) Integrating te continuit equation (.d) wit respect to, leads to u v w d d d x z. (7) As te lubricant film tickness xz, s bs as bs as, using te Leibniz integral rule f xs, f x, s dx dx f b s, s b s f a s, s a s s and appling te boundar conditions, for te continuit equation (7) follows u d u v wd x x z. (8) o 6

7 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 Integrating te first term of (8) wit respect to, (5) leads to and rendering into account u d q d kq d d kq d. (9) Analogousl, te integral of w in te fort term of (8) is obtained b (6) in a form wd q d kq d. () B using (4) te integrals of,, in (9) and () are respectivel equal to d ; d ; d. () 8 Ten after substitution of te velocit components (5), (6) in (8) and using () te following Renolds tpe equation is obtained 5 p p 5 p p r k k. () x x 8 x z z 8 z x Tis is te modified Renolds equation for a finite lengt journal bearing lubricated wit non-newtonian Rabinowitsc fluid under stead state conditions. Te tpe of equation coincides wit te one obtained in [] as ere te coefficient of pseudoplasticit k is also involved. It is known tat for Newtonian fluids k and from () can be obtained te classical Renolds equation for Newtonian fluids. Te above equation is more general tan classical one, as it includes te case of non-newtonian lubricants ( k ). Tis equation as to be represented in a non-dimensional form b introducing te coordinate transformations and corresponding dimensionless variables, according to substitutions: x x z ; z ; R r L r ; L c ; r c p ; 6Ur H, () c were: - circumferential coordinate, z - dimensionless axial coordinate, L - bearing axial lengt, - diameter to lengt ratio, - clearance ratio, - dimensionless pressure, U r - journal circumferential velocit, H - dimensionless film tickness. All variables in () are canged wit non-dimensional ones (), as for te partial derivatives te following equalities are in effect x r p 6r x c ; ; z L z ; p 6r z z L c ; (4) 7

8 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/8457 c H. x r After substitution of () and (4) in () and appling some matematical transformations te final dimensionless form of D Renolds equation for pressure distribution is obtained as: 5 5 H 6H H 6H H 8 z z 8 z. (5) In tis equation te non-dimensional nonlinear factor, wic takes into account te non-newtonian beaviour of a fluid, described b te cubic model, is involved. According to [,, 8, etc.] te nonlinear factor is represented b te following relationsip U r k k c c. (6) Wen te nonlinear factor is set to zero, te Renolds classical equation for a journal bearing lubricated wit a Newtonian fluid is obtained. At pseudoplastic non-newtonian fluids, wile at te dilatant ones [, 6, 8]. As it can be seen from (6), te nonlinear factor values depend on te coefficient of pseudoplasticit, te lubricant initial viscosit, te velocit of journal, and te radial clearance. Given te relation (6) between te nonlinear factor and te coefficient of pseudoplasticit, te increase of k increases te values of. For Newtonian fluids because k. Namel b introducing tis dimensionless variable into equation (5), te reological caracteristics of te pseudoplastic and dilatant fluids, wic can be described b te Rabinowitsc fluid model, are taken into consideration in te matematical model. Particular cases As it was written in [], te analsis of Renolds tpe equation as () can be performed for small and respectivel for large values of te nonlinear factor b using different approaces. Furtermore, for small nonlinear factor two particular cases of D journal bearings lubricated wit a Rabinowitsc fluid can be considered at using te long and sort bearing approximations, wic are b te oter side extensivel applied in te studies of lubrication teor. It is well known tat for an infinitel long bearing te derivative p z, because of wic te modified Renolds equation for suc kind of bearings is obtained from () and written as 5 p p r k x x 8 x x. (7) Using () and (4) a non-dimensional form of (7) satisfies te equation: 8

9 MATEC Web of Conferences 45, 7 (8) NCTAM 7 ttps://doi.org/.5/matecconf/ H 6H H 8. (8) Te pressure distribution for te case of an infinitel sort bearing is governed b te following equation, wic is also a reduced form of (): 5 p p r k z z 8 z x (9) Here te second term on te left and side of () is vanised since p x p z. Dimensionless form of (9) is obtained b using of () and (4) and is given as: 5 H 6H H z z 8 z () Bot of te equations for infinitel long and sort bearings, respectivel, can be solved b analtical or numerical metods as several solutions of similar tpe of equations are alread publised. 4 Conclusions Tis paper presents precisel, step b step, wit necessar explanation obtaining of a modified Renolds equation for lubrication of finite lengt journal bearing lubricated wit non-newtonian Rabinowitsc fluid. Starting from te momentum and continuit equations, and based on te cubic stress constitutive equation, te velocit components for two dimensional flow of a Rabinowitsc fluid are obtained. For tis purpose differentiation and integration tecniques are applied. Te constitutive equations for te sear stress and sear strain rate are represented b a cubic function of a special tpe, wic finall, after matematical treatment, are reduced to semi-cubic equations for teir components. An approximation to te real root of semi-cubic equation is used suc as in []. It is possible to find b Cardano s metod all roots of te semi-cubic equation, and in furter matematical processing to operate wit some appropriate teir approximations (due to te complexit of Cardano s formula). Afterwards, a standard approac for te lubrication teor is applied. Integrating te continuit equation across te fluid film and substituting te obtained velocit components in it, te nonlinear modified D Renolds tpe equation is derived. Te obtained equation is sligtl differing in coefficients from tose in []. Tis equation is in te appropriate form for numerical modelling and solving b some mes metods or adapted computer codes ones. Based on te presented above, a stud of te dilatant and pseudoplastic nature of te lubricant in comparison wit Newtonian fluid can be carried out. Te performances of journal bearings lubricated wit a non-newtonian Rabinowitsc fluid can be compared wit te case of Newtonian lubricant troug te variation of te non-newtonian parameter, i.e. te nonlinear factor. Te future analses can be performed wit combinations of oter effects in lubrication performances of journal bearings as influence of fluid inertia, elastic deformations effects, surface rougness, etc. 9

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