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2ndOrder - Non-Newtonian Lubrication With W G Sawyer...

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Unformatted text preview: Non-Newtonian Lubrication With W. G. Sawyer Graduate Student. J. A. Tichy Professor. the Second-Order Fluid In certain applications where the lubricant is subjected to rapidly changing conditions along its flowing path ( such as an elastohydrodynamic contact ), the inherently time dependent nature of the lubricant may be significant. The simplest type of model to correctly account for such time dependence is the second-order fluid, which is a Rensselaer Polytechnic Institute. Department at Mechanical Engineering, Aeronautical Engineering. and Mechanics, Troy NY 12180-3590 systematic small departure from Newtonian behavior, involving higher order rate- of—rate-of strain tensors. As in a companion paper using the Maxwell model, the formalities of applying such a model to thin filmflow are emphasized. Using a regular perturbation in the Deborah number, with the conventional lubrication solution as the leading term, a solution can be obtained. Viscoelasticity may raise or lower pressure depending on the nature of edge boundary conditions. Introduction The present paper is a companion to a recently published study of the so-called upper convected Maxwell model in lubri- cation. Tichy (1996). The purposes of that paper were three- fold: 1) describe the conditions under which a fluid time depen- dent response occurs (purely viscous materials respond instanta- neously). 2) present the mathematical formalities which are necessary to properly account for the time dependence (admissi- bility or objectivity). and 3) illustrate some typical behavior which may occur in simple contact geometries. Fluids which exhibit such a time dependent response are viscoelastic. There are similarities and differences betWeen the behavior of the second-order fluid considered here and the Maxwell fluid. The main similarity is that both models account for time depen- dent fluid behavior. Consider the flow of an inertialess fluid confined between infinite parallel plates. which experiences a sudden inception of shearing motion. The Newtonian fluid (stress proportional to rate-of-strain) responds with an equally sudden rise in shear stress. Both the Maxwell and second-order fluids respond with a delayed rise in shear stress as it approaches the steadyostate value. The time constant of the response is A... a characteristic material time parameter (relaxation time). One difference between these two viscoelastic models is that in the Maxwell fluid, the time response is introduced through a rate-of-stress term; while for the second-order fluid, the time response arises due to a rate~ofvrate~of-strain term. The second- order fluid model is thus explicit in stress; while for the Maxwell fluid. a coupled differential equation must be solved for the stress. The second order fluid is therefore easier to apply. A second difference is their conceptual origin. The Maxwell model arises from idealizing neighboring fluid particles as connected by an embedded differential position vector of length ds. The stretching of ds causes stress between the adjoining particles, as if induced by a viscous damper (dashpot) in series with a spring. The second-order fluid. however, accounts systemati— cally for small deviations from Newtonian behavior. The Max- . well fluid can account for large elastic effects. but relies on a specific empirical description of the behavior (springs and dashpots). The second order fluids can account, in theory, for only vanishingly small elastic effects. However, no specific idealizations are required—all proper time dependent models asymptotically reduce to the second order fluid in the limit of small ho. Contributed by the Tribology Division for publication in the JOURNAL or: Tntnotoor. Manuscript received by the Tribology Division February 7. 1997; revised manuscript received June 30. 1997. Associate Technical Editor: C. Cusano. 622 I Vol. 120. JULY 1998 Copyright 9 1998 by ASME As discussed in the earlier paper. terms such as “lubricant rheology," or “non-Newtonian lubrication" generally refer only to the effect of variable viscosity. a small subset of rheolog- ical behavior. In the one-dimensional lubrication flow. where 6111/ dx. é omen, this so-called purely viscous or generalized Newtonian model becomes: a ~ 81" 2| = — v fix; The subscripts 1, 2, and 3 represent the fllmwise. crosswise, and spanwise directions of a cartesian basis, respectively. For thin film flows in two dimensions. we have. ’ a... was 7'21 = “Fl‘l'zl- 77:1 ‘3 a - 7:3 — 3x 2 2 l1 = M). r = M. + vb. i.e.. to satisfy invariance considerations, viscosity must be seen as a function of the magnitude of the strain rate (the second invariant of the strain rate tensor). The Bird et al. (1987) con- vention for the sign of the stress is used— positive normal stress is compressive, as is pressure. In purely viscous fluids. the stress must always be precisely in phase with the applied motion. However, it is known that additive-containing lubricants possess a characteristic (relax- ation) time A.) which is of order 10“ to 10 ‘65. If a lubrication process time (in the convective sense) is of this order. we can expect strong time dependent effects. i.e.. the fluid cannot respond to imposed conditions. Such conditions may occur in an elastohydrodynamic line contact. i.e.. a lubricant element experiences changes on the order 0.1 ms as it flows through the contact. The Deborah number De = hoU/L is a measure of such time dependence. For Newtonian quasi-steady behavior Deborah number is zero. and the second-order fluid model ap— plies for De e 1. For homogeneous flow. such as simple shear. where all fluid elements or particles experience the same kinematic conditions. the second order fluid model for the shear stress is. 31) 5 3L) T21=“l-‘o'-l+l\o#o—t( 1)- If fluid relaxation time )to is zero. the Newtonian model is recovered. The question arises as to how to change this homogeneous model for more general flows. The answer is that we must replace the time derivative either by one written with respect to coordinates which translate and rotate with the fluid particles 721: ‘fll’zts #- : Milan )- 7'23 = " #‘lr'zsq Transactions of the ASME (the eo-rotational or Jaumann derivative ), or coordinates which deform with the fluid (the codeformational or convected deriva- tive) . Background is presented in Bird's ( 1987) text. The deriv- ative in the corotational or codeformational system is then trans- formed back to the conventional spatial coordinates by the chain rule. Unfortunately. such derivatives greatly complicate prob- lems by introducing strong noniinearities into the equations. Using the partial derivative in the above equation is clearly not sufficient—it represents the time rate of change of rate-of-strain at a fixed location. A model so written refers to drficrent particles passing through a specific location. The conventional material or substantial derivative of fluid mechanics is also inadequate—it is the rate of change of a property of a given particle with respect to coordinates which translate with the particle but maintain their orientation (do not rotate or deform). However, an observer rotat- ing and translating with the flow will calculate a different stress state than one just translating with the flow. Only if a corotational or codeforrnational derivative is used will the resulting model be admissible, i.e., all observers will calculate the same stress state. The preference of one admissible derivative relative to another is based on the practical issue of which seems to best predict experi- mental results in normal stress behavior (they both predict the same shear stress behavior). The second-order fluid has several potentially fatal theoretical flaws, capable of producing ill-posed problems. if applied to finite Deborah number conditions, see Bourgin and Tichy (1989). Perhaps due to these problems, applying ordered fluids to tribologieal flows of time dependent materials seems to have fallen out of favor in recent years, in favor of the Maxwell fluid. Prior to the 1990's there were a number of studies of ordered fluids in plane creeping flows: Bourgin (1982), Tichy and Ku (1988), Rajagopal (1984), Fosdick and Rajagopal (1979), Tanner (1969). to cite a few. if the second-order fluid and the Maxwell fluid are both correctly applied to the same problem, the second-order fluid is much easier to use. With the ordered fluids (being explicit in stress), stress can be plugged into the equations of motion, producing differential equations for the velocities. With the Maxwell fluid, the stress differential equa~ tions are nonlinearly coupled to the equations of motion. Unfor- tunately, in most of the existing Maxwell-lubrication studies various ad hoe assumptions are made to dec0uple the stress, as discussed in the companion paper, Tichy (1996). Analysis We follow the notation of Bird et a1. (1987) and write the secondeorder fluid model in terms of the contravariant con- veeted derivative, ' Tu 3 ‘rlto(7u)tj " M‘hzm — hoo'l’rnrml’uw) Nomenclature Material property parameters are viscosity yo, and characteristic fluid times AG and 70° (relaxation time and retardation time. respectively). We use Cartesian tensor indicial notation with summation on repeated indices—the extra stress tensor is r0. The rate of strain tensor is defined as, 6v. 'l’mrj = '3"; = (Vi-Ur} + (V1015. (V0.1: :3": - (7-) The superscript Tdenotes the transpose: (. . . 5 = (. . .),—,—. Note that the velocity gradient convention is different than that often used. The subscript (1) indicates the first convected contravari- ant derivative of the strain, i.e., the rate-of—strain. The subscript (2) denotes the second convected contravariant derivative of the strain, i.e., the rate-of-rate-of—strain: D . Var-'1‘ = E 'l’ij ‘ 7rm(vv)mj — (VVHquj‘ _ D 8 6 —-i'=--i‘+um—-i'n 3 r): 7’ at 7’ ax, 7’ ( ) where D/Dr denotes the usual material derivative which follows the particle. The equation of continuity for incompressible flow is con- cisely written as (4) while the momentum balance equation for inertialess flow with- out body forces is, 67L,“ 6x... = 0, 7n,- = 7,, + p6,, (5) The total stress tensor is m}, which is composed of the extra stress plus the isotropic pressure p, where 6,,- is the Kroneker delta. In most flows of Newtonian fluids. the total normal stress (defined with Bird's convention) approximately equals the pres- sure. We now write out these equations in component form assum- ing two-dimensional flow with v.(x,, x;, t) and v2(x,, x2, I), see Fig. 1. We obtain the continuity equation: a = (—) slope parameter = (Hm. — Hexit)/Hinletv Eq- (23), Fig. 2 De = (—) Deborah number = hoUl L r H, 1-1., = (m) film thichness, reference value. Fig. l k,” k, = (—) integration constants, Eq. (11) L = (m) contact length parameter, Fig. 1 p, p, = (Pa) pressure, ambient pres- sure, R = (m) journal bearing radius. Fig. 2 t = (s) time, U = (m/s) sliding velocity, Fig. 1 Journal 01 Tribology 81:, av; —- + — = 6 (l) at, 6x2 ( ) v., v; = (m/s) velocity components #0 = (Pa — s) viscosity (filmwise, crosswise), Fig. 1 1r... r“, = (Pa) total, extra stress com- ): = (—) filmwise dimensionless coordinate, Fig. 1, Eq. (21) x1, x2 = (m) spatial coordinate com- ponents (filmwise, cross- wise), Fig. l '51,], 7“,»,- = (ll 5) rate-of-strain compo- nents 6,7 = (—) Kronecker delta 6 = (— ) eccentricity ratio, Fig. 2, Eq. (23) 9 = (-)journal bearing filmwise coordinate, Fig. 2, Eq. (23) AD, hm = (s) fluid relaxation, retarda- tion time parameters ponents, 1r”. 1'“, = (—) total, extra stress compo- nents (dimensionless), Eq. (21) w = (1/ s) journal bearing rota- tional speed, Fig. 2, Eq. (23) Superscripts [N] = denotes the Newtonian (lubrication) theory solution {D} = denotes the Deborah number (viscoelastic) correction solution JULY 1998, Vol. 120 l 623 Fig. 1 Schematic ct arbitrary one-dimensional contact geometry and the inertialess (creeping flow) momentum equations in the absence of body forces: 6—1) +6—7” + 6T2] 0: 6p +37” + SE _ 5x1+6x1 ax; 6X2+ 3x, 6x; If we now use the lubrication (thin film) scalings (u2 < v]. din/ax. <5 Bur/3x1,3v3/6x. < avzlaxz), we obtain the following expressions for the extra stresses, retaining the largest terms multiplying the material parameters: 0= (7) 6U 6v 2 Tu = ‘21-10 6x; — 2H0M(Bx:) 8U: 2 321’] + 2 A + 2 A “0 00(Bx3) “0 ”ans: 7' —2 av-2A ua—ZU,‘+U alu‘+2(§—‘-u-'—)2 22 #0 6x2 #0 0 I 8x; 2 Bxlaxz 6x2 301 2 3v: + 2 A --- - 2 it #0 oo( c912) #0 08x1 31’ av; 821:123—2v1 , = = — +2 A. +1} 7"] T1: ”0— 8x2 #0 0|}! anon 28x5 at] 2 5m + 2(6x;) ] + [Johnaxzar- (8) Cross differentiating the pressure gradient terms of Eq. (7), and substituting the extra stress terms of (8) gives as the governing equation: 0 = MOB—LL3 IMK0(U1— 641)] 64Ul+ (94 U| )- + - 9 ”’67; Bxfixi ”1 ax; 6x36: ( ) The classical (Newtonian) lubrication governing equation is obtained if )to = 0. Note that the retardation time Am disappears from the formulation. Clearly, by inspection, the solution to the Newtonian case aha/(kg = 0, also satisfies the viscoelastic case of Eq. (9), consistent with the plane flow theorem of Tanner [1966}. The boundary conditions for steady flow from Fig. l are, x2=0:u. = U,v2=0; x2=H(xl):u1=uz=0 (10) The solution to Eqs. (9) and (10) can be written as H x§ x I: u. = U|:(ka+kb7i-9)(-fi—z—Ez) + r «3], (II) where k.l and kb are integration constants from Eq. (9). Differ- entiating (ll) with respect to .r. , substituting this expression in 624 I Vol. 120, JULY 1998 the continuity equation (6). integrating the result with respect to x;, and applying the a; boundary conditions of (10) gives: dH 2 H0 xi x3 = —-— — +k- ‘——~ , kg: 1); de1(3k .H)(H3 H2) 3 (12) These velocity field results are exactly the same as would be ' obtained for the Newtonian fluid. The remaining constant must be found from the edge bound- ary conditions on the normal stress, analogous to the pressure boundary conditions in the Newtonian case. Herein lies contro- versy. The equations of motion (7) can be identically expressed in the following form: 6722 6721+ 1032 _ 711) 51722 _ _ a 6x1 BX}; BX] ax: — 8x1 . (13) Substituting for the r.)- from Eq. (8), and comparing orders of magnitude from the lubrication velocity and length scalings. it is easy to see that Brag/8x1 < 6122/61.. i.e.. the load carrying normal stress «22 does not vary across the contact, just as the pressure does not vary in the Newtonian case. Thus we find. dTr’zz l H0 =2 U— +k dx. ’1” 112(3Jr ”H) 3 l (#11 H0 H0 —2 AU'—-— 2+3k—+k , I4 #0:) dx ( bH 311:) () where the right-hand side is known, and the load carrying stress is fizzixr) = ”2(0) +L’Id -—:=(xl)de (15) If the edge condition n22(L) is specified, then Eq. (15) leads to a quadratic equation for kb which can be readily solved. The normal stress in the flow direction can be found from: ”HUM 12) = ”22(11)‘ [7'22 “ Tarn-T1— x2). dH 1 H 3 7'22“ 7'” = "4}L0U2x'jfi(2 +19%) (3%- 2%) 2 +2yvoU2%[<3+kb%)(2:—;—l)—1]. (16) Note that the normal stress it” varies across the gap. i.e.. in the crosswise x; direction. Similarly, substituting from Eq. (8) and evaluating conditions at x; : 0. the shear stress at the lower surfaces, which produces global friction. is given by, I-toU (4H0 H43) Ho Ho H H- 1 (11"! H5 H3) +2 )toUz—— 2 _+k — . 17 #0 Lder( bH, ( ) 7'21 = The controversy has to do with specifications of the edge boundary conditions. In viscoelastic flows “pressure" has no direct physical meaning. For incompressible flow, the total stress in,- exerts the physical force, the extra stress in, is the portion of the total stress which can be determined by the flow kinematics and the rheological model. and the pressure p is the difference between the two, as can be seen from Eq. (5). In Transactions of the ASME the case of periodic boundary conditions, such as those required in a journal bearing analysis. one possibility is as follows: (a)1r22(xl) = Trzztxi + 2r). (“7112(0) = 7T2:(27FR) = Pa- (18) These conditions produce a stress profile similar in form to the pressure profile of Newtonian lubrication. However, in the absence of some specific feeding arrangement, there is no physi- cal reason to support setting the normai stress #22 to ambient at the maximum film thickness point. 0 = 0. As an alternative, the following boundary conditions are pro- posed: (a) 7r22(xl) = "22”: + 27)» ZKR (b) O :f (“‘22 _ pa)dxls 0 (19) i.e,, a periodicity and a normalization condition. The normaliza- tion condition (b) would seemingly apply in a submerged con- figuration—the bearing cannot build up a net stress relative to the ambient pressure. Such conditions are commonly used in the case of lubrication with inertia. Tichy (1987). In the New- tonian case. these two conditions (18) and ( 19) are equivalent. We are not addressing here the issues of cavitation (see discus- sion below), but half-Sommerfeld-type boundary conditions could easily be used by discarding the region of negative normal stress. In the case of an open ended contact such as the slider bearing none of the boundary condition alternatives is entirely satisfac- tory. The following case is “clean.” i.e.. does not require aver- aging. and is often used: «22(0) = 7722(L) = Pu- (20) However, there is no obvious physical argument in its favor. Further, there is no reason at all to set film pressure (which for viscoelastic fluids varies across the gap) equal to ambient pressure at the film ends. Unfortunately, this has been done in many studies, which clouds the complex issues concerning time dependent fluid effects in lubrication, see the original discussion of Tanner( 1969). To our knowledge. there are no experimental results which directly pertain to this issue. The best alternative would seem to be that the flow directed normal stress 7r” must balance the ambient pressure at the film ends. which would be the situation in the case of a stationary free surface with negligible surface tension. However, 1r“ varies across the gap, so the best we can do is. ’h'u(0) = find) = Pa. 77110“) = 1 fl ”(10L WIN-Yb Izldxz ,1 H0 1 = ,, +2 )tU-— 4+2k—+— 7r--(xi) #00 H1( ab, 3 ,H2 of?) (21) We wish to present our results in dimensionless form and thus introduce the quantities: XI ”(xl) )KoU = ~— ' h = y D = —— x L “0 Ho 6 L “=(Wti—p03' w=(7r22—pa3’ T”: Tll . (22) uoULlHa " ,uoUL/Ha yoU/Ho Using these dimensionless forms, from Eq. (19), the normal stresses are related as, 1 l i l Knot) = 7r”(x) + 2 De F (4 + 2kg; + '3- k}, P) (23) Journal of Trihology For demonstrative purposes, we analyze two specific film shapes, the journal bearing and the simple wedge contact (plane slider). respectively: h=l+ec050, 0=277x; h=1—a.x, (24) see Fig. 2. For the journal bearing case, H0 is the radial clearance and the sliding speed U = wR. For the wedge case, He is the inlet film thickness. For the journal beating, perfon'ning the integrations of Eq. (15) and nondimensionalizing, we have: mm) = «”(0) + 6130(6; e) + k,,I‘§°(6; e) l l 1 1 ‘Deiziu+e)=”F)+2k“(u+e)’_P) 1 l ((l+c)‘_E)i (25) 1~(arccos(€+cosa)). Osdsrr l—e‘ h 1 + [$027—-2(2n—arccos(f———L59)), n56527r k Wu 1 +- 2 Vl-e h m_ l (—esin9 10°) 2 (1—53) h ‘ ' m 1 (~esm8 00 00) = , j 3]» ~f 26 3 2(1~5-) h‘ ' ‘ ( ) h-Hu(1+ean50) It —— ZRR °l Fig. 2 One-dimensional contact geometries tor (anoumal bearing (peri- odic contact), and (b) inclined plane slider bearings (open ended wedge contact) JULY 1998, Vol. 120 I 625 1,, (y=0) 1 1 . l 1 +;(3+kb;) -2Deestn0(2;1-2+kbfi) (27) l h For the wedge contact case, we have: 1r,,(x)=rr,.,(0)+-6-(I-%)+£(l-Fli) a a _Dea[2(1-é)+2k,(1-%) 12 _i +2k,(1 ’14)], (23) 1 1 1 1 1 Z+;(3+kb;)—2Dea(2P+ka) (29) We pick here several cases to demonstrate the solution tech- nique. In the journal bearing case, we find, h(0) = h(27r) = l + e. For the case when normal stress is ambient at the beginning of the film, Eq. (18), and Eqs. (25)-—(26), we find. increasing eccentricity a = 0.8. r. = 0.5. E = 0.2 (Newtonian) I! load carrying normal strut x 0 W2 I 3N2 2n angular coordinate along filmo Fig. 3(a) Newtonian com...
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