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Unformatted text preview: NonNewtonian Lubrication With W. G. Sawyer Graduate Student. J. A. Tichy Professor. the SecondOrder 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 secondorder ﬂuid, which is a Rensselaer Polytechnic Institute.
Department at Mechanical Engineering,
Aeronautical Engineering. and Mechanics,
Troy NY 121803590 systematic small departure from Newtonian behavior, involving higher order rate
of—rateof 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 socalled 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 secondorder 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 rateofstrain) responds with an equally
sudden rise in shear stress. Both the Maxwell and secondorder
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 rateofstress term; while for the secondorder ﬂuid, the time
response arises due to a rate~ofvrate~ofstrain 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 secondorder 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 “nonNewtonian lubrication" generally refer
only to the effect of variable viscosity. a small subset of rheolog
ical behavior. In the onedimensional lubrication ﬂow. where
6111/ dx. é omen, this socalled 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
additivecontaining 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 quasisteady behavior
Deborah number is zero. and the secondorder 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 eorotational 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 rateofstrain
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 secondorder ﬂuid has several potentially fatal theoretical
ﬂaws, capable of producing illposed 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 secondorder ﬂuid and the
Maxwell ﬂuid are both correctly applied to the same problem,
the secondorder 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 Maxwelllubrication 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"; = (ViUr} + (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 rateof—strain. The subscript
(2) denotes the second convected contravariant derivative of
the strain, i.e., the rateofrateof—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 twodimensional 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, 11., = (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) rateofstrain 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 onedimensional 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 = ‘2110 6x; — 2H0M(Bx:)
8U: 2 321’]
+ 2 A + 2 A
“0 00(Bx3) “0 ”ans:
7' —2 av2A 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+kb7i9)(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 righthand 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 “ TarnT1— 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, ItoU (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 halfSommerfeldtype 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 WINYb 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 Vle h
m_ l (—esin9 10°)
2 (1—53) h ‘ '
m 1 (~esm8 00 00)
= , j 3]» ~f 26
3 2(1~5) h‘ ' ‘ ( ) hHu(1+ean50) It —— ZRR °l Fig. 2 Onedimensional 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;12+kbﬁ) (27) l
h
For the wedge contact case, we have: 1r,,(x)=rr,.,(0)+6(I%)+£(lFli) a a _Dea[2(1é)+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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