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Thermal elastohydrodynamic lubrication of rolling and sliding contacts with a power law fluid
77
Wear, 154 (1992) 77-93
Thermal elastohydrodynamic lubrication sliding contacts with a power law fluid Tsann-Rong
of rolling and
Lin
Department of Mechanical Manufacturing Engineering National Yun-Lin Instihde of Technology, Hu-Wei 63208 (Taiwan) (Received
April 24, 1991; accepted
June 7, 1991)
Abstract
A modified Reynolds equation for compressible fluids which obey the power law model was derived. Full numerical solutions were obtained by coupling the equation for lubrication with those for film shape and energy between two surfaces with simultaneous rolling and sliding motion. The effects of various power law indexes and slide-roll ratios on lubricant temperature, pressure, film thickness and friction coefficient were investigated. The results indicate that temperature effects are significant and cannot be neglected.
1. Introduction
Common lubricants exhibiting non-Newtonian behaviour include polymer-thickened oils, greases, and natural lubricating fluids which appear in animal joints. These lubricants violate the Newtonian postulate which assumes a linear relationship between shear stress and rate of shear. In recent years, a modified Reynolds equation for a power law fluid [l-5] has been derived. The power law model has received attention because of its simplicity and potential for describing many lubricants such as silicone fluids and polymer solution. The power law index n is essential in the study of power law fluid flow problems. For 12= 1, the fluid is Newtonian; for n < 1, the fluid exhibits pseudoplastic behaviour; for n > 1, the fluid is a dilatant. Dien and Elrod [l] examined non-Newtonian lubrication, and developed a perturbation expansion for velocity and pressure fields under Couette-dominated flow. Sinha and coworkers [2-4] analysed the isothermal non-Newtonian lubrication problem theoretically with rigid rolling disks under pure rolling conditions. Wanget al. [5] analysed the isothermal elastohydrodynamic lubrication problem using a power law model under pure rolling conditions. In these papers [2-51, for a power law model fluid, there are two regions (d&lx>0 and dpl dx < 0) with different Reynolds equations. The viscosity is assumed to vary exponentially with pressure. The solutions were limited to pure rolling conditions and incompressible fluids. For most practical lubricants, the viscosity and density depend strongly on temperature. For this reason, thermal effects must be considered, for they affect the film thickness and all other performance characteristics. Cheng and Sternlicht [6] developed a numerical scheme to handle the thermal Newtonian problems of elastohydrodynamic lubrication in rolling and sliding contacts. Ghosh and Hamrock [7] obtained a numerical solution for thermal Newtonian EHL of line contacts using the finite difference formulation. They analysed the film shape by subjecting each rectangular area to uniform pressure.
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78
Recently, Najji et al. [S] presented a generalized Reynolds equation for nonNewtonian thermal hydrodynamic lubrication. Yang and Wen [9] obtained a modified generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic or hydrodynamic problems, whether the film is in a steady or a transient state. Although the previous generalized Reynolds equations can incorporate most rheological laws found in the literature, they cannot include the power law model for rolling and sliding contact. In this paper, a thermal non-Newtonian fluid analysis including effects of pressure, film thickness, rise in temperature and coefficient of sliding friction in rolling and sliding contact, is presented. The isothermal non-Newtonian results obtained by Houpert and Hamrock [lo] were modified slightly to suit the power law model which is derived by a perturbation method [l], and the proper energy equation for thermal situations is considered. A mean temperature across the film in each section is calculated by an approximate method. Then the lubrication equation is coupled with equations for film shape and energy, and they are solved simultaneously by a numerical method. 2. Mathematical
analysis
In a lubricating system, the primary objective is to develop a lubricating oil film to support a moving body with a load on it. A thin film in an EHL problem is under such a high pressure that the density and viscosity would create substantial variations in a real application. The pressure is so high on a very narrow film that even liquid oil is characterized as a compressible fluid. In this study the numerical scheme developed by Houpert and Hamrock [lo] was revised to suit the features of power law fluids. The derivations and the criteria for computation are based on the following assumptions: (1) elastic deformation of the surface is calculated for a semi-infinite soiid under plane-strain ~nditions; (2) oil side-leakage is neglected owing to the relatively broad width in this direction; (3) the boundary conditions needed to depict the narrow region are P=O at a large distance from the high pressure zone, and P=aPbX=O at the cavity outlet. 2.1. Flow equation From an order of magnitude analysis, we obtain the Navier-Stokes equation which is consistent with the basic assumptions of creeping flow. These equations and their boundary conditions are given as (see Appendix A)
ar
ap -=a~
ay at y=O, u=u, at y-h, u=ub
In the one-dimensional power law model is n-1
au
II
r=rnay
au
ay=qay
case, the constitutive
equation
for the fluid satisfying the
*au
where q*=rn/au/ay~-' is usually called the equivalent viscosity of the fluid. m is the consistency coefficient and n is the power law exponent. The fluid will either have zero or infinite viscosity outside the contact for values of II other than unity. A modification of the power law model is proposed [ll]. At a low shear rate (y-0) the fluid is expected to behave as a Newtonian fluid. At a high shear rate (+-, a)
79
the lubricant becomes Newtonian again with the same viscosity as the base lubricant. These two cases are not considered here. Substituting eqn. (2) into eqn. (l), we obtain aP a -=-
ax
ay
au
( 1
(3)
‘7ay
The asymptotic expansion is introduced to denote all related properties in terms of the small dimensionless parameter E. The x-component velocity is therefore given in the form u=u*+cu~+...
(4)
The velocity gradient
is then approximated
as the asymptotic
form
or we can write
The apparent series
viscosity and fluid-film pressure
are expressed in the forms of asymptotic
p=po+q1+... Assuming
(6) Couette-dominated
flow, the pressure
gradient
can be written
as [l]
-ap =e- a+
ax
ax
By substituting eqns. (4) and (6) into eqn. (3) and by equating power of E, one obtains
a
* ( )
G q”
au0 -= ay
coefficients
of equal
0
(8) The solution for eqn. (7) associated y = h, u =ub, gives the velocity u. as (*b
-
uo= -y-t*, h
analysis of eqn. (8) gives a solution 1
kn[(&,-&)/hr-’
y-0,
u=u.
and
1
for the first-order
velocity u1
3 cv’ -yh) aX
Thus, the velocity with the limiting
* =
conditions
*a)
An analogous *I=
with the boundary
2
v_yh)+
%n[(ub - u,)lhr- l 3.X
condition b$dy+,,
that E is sufficiently
small is written
as
(11)
80
By substituting the velocity function into the continuity equation and assuming ph = p,h, where p = ap/&x= 0, the modified Reynolds equation for a power law fluid is given as
hn+2 ap _-+!i?$%)“(h_!!$=)
SRI-”
(12)
12mn ax
Equation n = 1. The (12). The lubricants,
(12) can be reduced to the Newtonian-based Reynolds equation by putting result for pure rolling conditions from ref. 5 cannot be obtained from eqn. slide-to-roll ratio, which is the parameter which includes non-Newtonian is defined as a function of the relative tangential velocities of two surfaces:
SR = 2(ub - u,)/(u, + z+,) 2.2. Film thickness The elastic deformation
6 at any point x on the surface
is expressed
as
XC,“d &=-&
p ln(x-x’)’ s xmin
or in dimensionless
dx’
(13)
form as
Xcnd &-L
P ln(X-X’)2dX’ 2lr s xlni.
- $ In
(14)
A more accurate form of dimensionless elastic deformation & at node i was derived by Houpert and Hamrock [lo] as a function of the pressure Pi and the influence coefficient D, $=
$Dijpi-a
In
(15)
j-l
The film shape between parabolic approximation. written as Hi=Ho+
7
two elastic bodies The dimensionless
in line contact film thickness
may be expressed by a at each node i is then
(16)
+ 2Di,P, j-l
2.3. Energy equation The temperature distribution within the lubricant film is determined using the energy equation with appropriate boundary conditions. The energy equation for linecontact problems, neglecting heat convection across the film and conduction along the film, can be expressed as (17) By introducing written as
the dimensionless
variable
to eqn. (17), the energy
equation
can be
n+l
(18)
81
where m&4, + u$ kT
Q2=
f
Q3=
II-1
+’
R
0 7;z
0
@.&! (U +%,) kfR2
=
Integrating eqn. (18) in they direction twice, we obtain a two-dimensional temperature distribution which varies along both thex and y directions. As for Cheng and Sternlicht’s first approximation [6], we take the mean values of consistency m, density of lubricant p and temperature 0 along they direction. A mean temperature across the film thickness is introduced to replace the local temperature and is defined by 1 e,,.,(x)=
6’
s
dY
0
The average form of the energy equation written as
1
+ : ‘@H’-”
in terms of the mean temperature
is thus
1 36(1-~HJ$J)2
(n+3)(n+2)
x{I[3(1g+ q3+ I[? -3(,- !!&)]I”“) 1 -QfiH’-“(n+3;(n+2) Xii[,(lEquation
e
=
$$)(ZY-I)+
?]I””
_
?.-Ql~2?5
+
(19) dPldX=0.
It can be shown as follows
&Q-l-n
(20)
2
From ref. 12, the non-dimensional given as
em= 1.0
dY
(19) is not valid for PH=&H,where
h+eb
m
36(1-r.5~H,/~2
atX=-co
boundary
conditions
for the energy equation
are
2.4. Sliding friction coefficient From eqn. (3) the shear stress is
and the friction
coefficient
due to the sliding per unit width is given by
i_Jm(y)^dir/w Also, by introducing
(22)
the non-dimensional
parameters,
f can be expressed as (23)
2.5. Numerical analysis for the Reynolds equation If the side-leakage effect is neglected, owing to the extremely small high pressure contact region, the dimensionless Reynolds equation for a power law fluid of rolling and sliding contacts can be written as
with u=
fi=
m0uF -
K=48U(~l8v+’
E’R”
!E
E
p’
m0
PO
and the thin film thickness Hi=H*+
u,+ub u=s 2
x: 2
in dimensionless
form gives
N + EDijPj j-l
It is well recognized that the consistency is dependent on fluid temperature and pressure [13]. A commonly accepted relationship which describes the viscosity-temperature-pressure dependence is the Barus equation (25) The density
pi in the thermal
fluid film is related
to the local pressure
and mean
83
temperature.
The proper
o.6x10-9Hpi jYii= l-!? 1 + 1.7 X lo-‘&Pi [
expression
at X=X,,
P=aP/tlX=O
at X=X,,
(26)
{l.O-$!cT,(Bd--1.0)) I
The appropriate boundary conditions, on the cavitation boundary, are P=O
is given as
including
the Reynolds
condition
generally
used
The Reynolds equation is solved by iteration using the Newton-Raphson algorithm. The unknowns at present include the Iocation of the outlet meniscus X, the number of nodes for which the film pressure exists N, the film thickness constant Ho, the value of cS,E&where dPldx=O, and the node pressure Pi O-2, 3, _.. N). If the superscripts n and o respectively denote the new and old values corresponding to two successive iterations, the unknowns can attain convergent solutions by the relations (6%&) Pin
= =
WW + [Wk.%.)1 z-y+ (APj)” j=2,
ffOn
=
HJ0+(AH,)"
The Newton-Raphson j (i=l, 2, . . . IV):
algorithm
3, . . . . N
(27)
results in a set of N algebraic
equations
for the nodes
(28)
vi -saq
n+2 )7
a[tdPldx)il
_K
- fii
_
arii,
H._
apj (
3Pj
Dij- Ache [
I
q/ii) apj )
Since an equivalent number of equations additional load condition is needed:
jgq(&Y =(AV
is required
to solve N+ 1 unknowns,
an
(2%
where the weighting factor Cj is derived from the inte~ation of pressure at each subregion [lo]. The pressure adopted in this study is described as a linear function. Thus,
84
a linear system of N+ 1 equations, for the solution of N-I-~ unknowns, is arranged in a compact matrix. All unknowns can converge to solutions if the correct initial values are used.
3. Computational
procedure
The arrangement needed to solve the thermal non-Newtonian elastohydrod~amic lubrication problems requires the isothermal cases to be solved first. The isothermal results are thus used as the initial conditions for further thermal consideration. The oil temperature is given by the mean temperature across the film. The temperature gradient in eqn. (19) is expressed in a forward difference form to be consistent with the isothermal case where the pressure gradient is also denoted by a forward scheme. The last term is integrated using Simpson’s integral method. Equation (21) can be integrated using the trapezoidal rule. The singularity at X=X’ is removed by assuming a linear function for the conduction term of the integration equation (21). Coupled solutions were considered to be convergent when the combined solution conserved flow from the inlet to the exit region in addition to satisfying the convergence criteria associated with various equations. The iterative method combined with an underrelaxation factor (the optimal value is 0.8495) can accelerate convergence for the temperature field. The iterative procedure is stopped to give the final temperature distribution when the errors in computing the temperature in the next iterative procedure become less than 0.0005. It takes 30-40 iterations to reach the convergent solution.
4. Results
and discussion
A modified Reynolds equation was derived which is based on a power law fluid with rolling and sliding contacts. This modified Reynolds equation is reduced to the Newtonian-based Reynolds equation if the power law index n is equal to unity. Table 1 and Figs. l-11 give the results for the power law model from which we can discuss the effects of the power law index n and sliding ratio SR on thermal EHL. The point of cavitation and minimum film thickness are given in Table 1 for various values of n and SR. For the compressible fluids, the point of cavitation Xend moves away from the centre-line of the contact as it increases. Prasad et al. [13] solved problems of compressible fluid film lubrication of rollers by power law lubricants. The tendency of the cavitation point to move, which is affected by the vaiue of n, is the same as that found by the author. Former researchers [Z-5] solved incompressible fluid problems in which the cavitation point moves towards the centre of the contact as n increases. The performance of a lubricating system with load W= 1.64 x 10e5, material parameter G = 3500 and velocity U= 1.0 X 10ml’, is fixed a$ constant values. The temperature increases as the power law index n increases for any specified rolling and sliding speed; this can be seen from Fig. 1. This result can be interpreted physically. An increase in n signifies an enhanced effective viscosity. Thus the resistance to motion would increase, leading to a higher viscous dissipation. The case of pure rolling conditions is not considered here, because the temperature rise is small in pure rolling cases, and the maximum temperature rise for the mid-film temperature occurs on the far left at the beginning of the inlet zone [7]. The variation of pressure under thermal non-Newtonian conditions is shown in Fig. 2. As the power law index n increases, the pressure increases, and the pressure spike moves towards the centre of contact.
85 TABLE
1
Numerical
results
n
SR
X end
h&R
0.9 1.0 1.1 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.0 1.0 1.0 0.9 0.9 0.9
1.2 1.2 1.2 1.2 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.5 1.0 1.5 0.5 1.0 1.5
1.10199 1.20611 1.29878 1.41375 1.37016 1.38617 1.39461 1.40722 1.26619 1.27850 1.28318 1.29452 1.19107 1.20413 1.20951 1.10116 1.10169 1.10235
7.434X lo+ 17.817x 1O-6 32.927 x 1o-6 59.141 x 10-6 54.539x10-6 56.530x lo+ 57.782 x lo+ 58.599x lo+ 32.118x1O-6 32.639x lO+ 32.878 x lo+ 32.958 x lo+ 18.513 x lo+ 18.046X lo+ 16.958 x lo+ 8.187x 1O-6 7.613 x 1O-6 7.163 x 1O-6
-4
-2
0
1.044 x lo-’ 2.469 x 1O-5 4.456 x lO+ 7.747 x 10-S 6.990x10-’ 7.279 x 10-T 7.480 x 10-s 7.638 x lo-’ 4.212 x lo-’ 4.307 x 10-5 4.372x lo-’ 4.420 x lo-’ 2.483 x 1O-5 2.474 x lo-’ 2.460 x lo-’ 1.117x 10-S 1.059 x 10-S 1.026x 1O-5
0.178 0.427 0.789 1.417 1.307 1.354 1.384 1.404 0.770 0.782 0.788 0.790 0.444 0.432 0.406 0.196 0.182 0.172
0.026 0.037 0.049 0.058 0.035 0.046 0.053 0.056 0.026 0.036 0.042 0.047 0.022 0.034 0.039 0.015 0.024 0.028
2
X-coordinate
Fig. 1. Temperature profile vs. different power law indexes from thermal non-Newtonian analysis. Non-dimensional parameters are W= 1.64X 10v5, G=3500, U= 1.0X lo-**, SR- 1.2: curve 1, n = 0.9;curve 2, n = 1.0, curve 3, n = 1.1; curve 4, n = 1.2.
86
i
5
0.0
$ 0.7 ce 0.6
-4
-2
0
2
X-roordinate
Fig. 2. Pressure profile W. different power law indexes from thermal non-Newtonian analysis. Non-dimensional parameters are W= 1.64X lo-‘, G=35OO, U= 1.0X lo-“, SR= 1.2: curve 1, n =0.9; curve 2, n = 1.0; curve 3, n = 1.1; curve 4, n = 1.2.
Then we find that the location
of the pressure
spike is the same as that of the
temperature spike because viscosities and densities are taken to be functions of pressure and temperature. The minimum film thickness becomes thicker as the power law index n is increased, which can be seen in Fig. 3. Figures 4-6 show that the higher the slide-to-roll ratio SR, the higher the temperature distribution for the three cases n = 1.2, n = 1.0 and n =0.9. The generation of heat in the sliding case due to the shear strain rate is greatly increased in the contact region, but is not significant in the inlet region. Owing to the effect of shear strain rate on film thickness, the lub~cating system of the non-Newtonian lubricants causes either thinner or thicker fihns. A power law lubricant results in both shear thinning and shear thickening effects. When n> 1, as the slide-to-roll ratio SR rises, the equivalent viscosity increases. This is the shear thickening effect. When nor; 1, as the slide-to-roll ratio SR increases, the equivalent viscosity decreases. This is the shear thinning effect. As the slide-to-roll ratio SR increases, the rnin~~ film thickness becomes thinner when nzzl, and conversely becomes thicker when n > 1, as shown in Figs. 7-9. When n 2 1, the behaviour of the minimum film thickness is the same as that of viscoelastic-plastic non-Newtonian fluids. The amount of oil flow increases relatively with the increase in power law index n. This phenomenon is explained by the increase in film thickness at a high power law index n. Figure 10 shows the mass flow rates for different power law indexes IZ. From Table 1 we see that the mass flow rate decreases as the slide-to-roll ratio SR increases when IZ5 1, but it increases when n > 1. The sliding friction coefficient values at different slide-to-roll ratios obtained by isothermal non-Newtonian analysis (dashed line) and thermal non-Newtonian analysis (solid line) are shown in Fig. 11. According to the thermal non-Newtonian analysis, the friction coefficients decrease with increasing
X-coordinate
-6.4
0.4
Od
Ii?
1.6
-4
-2 X-roordi
nate
parameters
-6.6
Fig. 4. Temperature profile 21s.different slide-ta-roll ratios from thermal non-Newtonian anaIysis. Non-dimensional G=3500, U=1.0~10-‘~, n=12 curve 1, SR=O& curve 2, SR=0.9; curve 3, SR=1.2.
-1.2
parameters
-Id
6.2 0.t
1.2 1.1 I 0.9 0.6 0.7 0.6 0.5 0.4 6.2
lb 1.7 1.6 1s 1.4 I.3
Fig. 3. Film thickness profile ~8. different power law indexes from thermal non-Newtonian analysis. Non-dimensional G=3500, U=l.OXlO-“, SR=1.2: curve 1, n=O.9; curve 2, n=l.O; curve 3, n=l,l; curve 4, n=1.2,
E *G
*?. 5
z v: 8 ”
it1 2 1.9
u
13
2
are W= 1.64 X IO-‘,
are WY=1.64 X 10e5,
0
-0.5 i.5
45
25
X-coordinate
-1.5
-0.5
parameters
0.5
Fig. 6. Temperature profile ~1s.different slide-to-roll ratios from thermal non-Newtonian analysis. Non-dimensional G=3500, U= 1.0x lo-“, n =0.9: curve I, SR =0.5; curve 2, SR= 1.0; curve 3, SR= 1.5.
X-ccmrd i na t.r
-1.3
parameters
-2.5
Fig, 5. Temperature profile vs. different slide-to-roll ratios from thermal Newtonian analysis. Non-dimensional G=3500, U=l.OXIU-ll, n=l.O: curve 1, SR=0,8; curve 2, SR=1.2; curve 3, SR=l.S+
-a5
t.5
are W= 1.64 X
10e5,
are W= 1.64 X 10--5,
0.5
,
,
,
,
,
,
0
,
X-coordinate
-6.4
, 0.4
,
, 6.6
,
, 12
, 1.6 -12
-0.6
-6.4
X-coordinate
0
parameters
-0.6
Fig. 8. Film thickness profile ‘us. different slide-to-roll ratios from thermal Newtonian analysis. Non-dimensional G=3500, U-1.0X10-“, n=l.O: curve 1, SR=O.B; curve 2, SR=1.2; curve 3, SR=1.5.
,
DA2-
0.44-
0.46-
0.46-
0.1-
0.52-
0.54-
0.56-
parameters
-12
E .+ Lr
0.56-
0,6-
O&z-
0.64-
0.66-
0.66-
0.7-
0.72-
0.747
ratios from thermal non-Newtonian analysis. Non-dimensional curve 2, SR=0.9; curve 3, SR=1.2.
,
I
Fig. 7. Film thickness profiIe vs. different slide-to-roll G=3500, U=l.OXlO-“, n=1.2: curve 1, SR=0.6;
-1.6
1.3:
1.4-
1.9-
1.6-
1.7-
1.6-
1.9-
2-
L1 -
22-
2.2 -
0.6
12
are W=L64X
10a5,
are W= 1.64X 10m5,
0.4
90
c
91 0.16 a17 0.16 2 c t
0.15 0.14
1;
0.13
2
0.12
5 E
0.11 0.1
;,
0.69
.F
a06
r=
0.67
b! 2 .L
0.06
2
0.66
a# 0.03 a02 aof 0 0
6.2
a4
a6
slide-roll
a6
1
ratio
SR
1.2
i.4
Fig. 11. Sliding friction coefficient for different power law indexes: curves 1, n=0.9; curves 2, n = 1.0, curves 3, n=l.l; curves 4, n= 1.2. The dashed lies show the results of isothermal analysis and the solid lines show the results of thermal analysis.
slide-to-roll ratio. Also, the friction coefficients remain constant under a high slideto-roll ratio SR for all power law indexes n. The low shear rate viscosity at high temperatures is the main cause of the decrease in friction coefficient.
5. Conchsions A numerical solution for the problem of thermal non-Newtonian elastohydrodynamic lubrication of rolling and sliding contacts has been developed. It calls for simultaneous solution of the thermal modified Reynolds equation, the elasticity equation, the energy equation, and the transient heat conduction equation in the solids. A compressible power law fluid is used to give more extensive and more believable results. The consistency and density are assumed to vary with temperature and pressure. The normal perturbation method is used to derive lubrication equations for the nonNewtonian power law model. By reviewing the procedures described in the analysis, thermal non-Newtonian power law model EHL problems are dominated by the power law index n and slide-to-roll ratio SR. The results obtained from this paper are as follows. (1) For compressible fluids, the location of the cavitation point moves away from the centre-line of contact as II increases. (2) As the power law index IZ increases, the temperature peak and pressure peak increase, and they move towards the centre of contact. (3) As the slide-to-roll ratio SR increases, the temperature increases and the minimum film thickness decreases when n IS:1, and conversely increases when n > 1.
92 (4) As the slide-to-roll ratio SR increases, the mass flow rate decreases when n < 1, but increases when it > 1. (5) Thermal effects reduce the friction coefficient as the slide-to-roll ratio is increased for all power law indexes n.
References
6 7 8 9 10 11 12 13
I. K. Dien and H. G. Elrod, A generalized steady state Reynolds equation for non-Newtonian fluids, with application to journal bearing, ASME, J. Lubr. Technol., IO5 (1983) 385-390. P. Sinha and C. Singh, Lubrication of a cylinder on a plane with a non-Newtonian fluid considering cavitation, Trans. ASME J. Lubr. Technol., 104 (1982) X8-172. P. Sinha and A. Raj, Exponential viscosity variation in the non-Newtonian lubrication of rollers considering cavitation, Weur, 87 (1983) 29-38. P. Sinha, J. B. Shukla, K. R. Prasad and C. Singh, Non-Newtonian power law lubrication of lightly loaded cylinders with normal and rolling motion, Wear, 89 (3) (1983) 313-322. S. H. Wang, D. Y. Hua and H. H. Zhang, A full numerical EHL solution for line contacts under pure rolling condition with a non-Newtonian rheological model, ASME J. Tribal., 110 (1988) 583-586. H. S. Cheng and B. Sternlicht, A numerical solution for the pressure, temperature, and film thickness between two infinitely long, lubricated rolling and sliding cylinders, under heavy loads, J. Basic Eng., (1965) 695-707. M. K. Ghosh and B. J. Hamrock, Thermal elastohydrodynamic lubrication of line contacts, ASLE Trans., 28 (2) (1985) 159-171. B. Najji, B. Bou-Said and D. Berthe, New formulation for lubrication with non-Newtonian fluids, ASME J. TtiL, III (1989) D-34. P. Yang and S. Wen, A generalized Reynolds equation for non-Newtonian thermal elastohydrod~~ic lubrication, ASME J. T&w& 112 (1990) 631-636. L. G. Houpert and B. J. Hamrock, Fast approach for calculating film thickness and pressures in elastohydrodynamically lubricated contacts at high loads, ASME J. TriboI, 108 (1986) 41 l-420. 0. Isaksson, Numerical analysis of elastohydrodynamic contacts using power-law lubricant with special reference to water-based hydraulic fluids, ASLE Trans., 30 (4) (1987) 501-507. J. C. Jaeger, Moving sources of heat and temperature at sliding contacts, J. Proc. R. Sot. N.S. W, 76 (1942) 283-224. D. Prasad, P. Singh and P. Sinha, Non-uniform temperature in non-Newtonian compressor fluid film lubrication of rollers, ASME J. Tribal., 110 (1988) 653658.
Appendix A: Nomencia~~ b c c,,
ci Dij
cb
half hertzian length (m), b=R(8Ww)‘R specific heat of lubricant (J kg-’ K-‘) specific heat of roller a and b (J kg-’ K-‘) weighting factors used to integrate P influence coefficient used to calculate elastic deformation at node i due to pi
E’
equivalent Young’s modulus (Pa), f
=
l--y;2 + Eb
G h H
materials parameter, G = CYE’ film thickness (m) dimensionle~ film thickness, H = hlctlb’
93
Ht? HO k kmkb K
4i m MO
rii n
N P pd” 4
Q
R SR T To US u u,,
@b
22 u
W x X X end -Knin Y
dimensionless film thickness where dP/dX=O dimensionless constant used in calculation of H thermal conductivity of lubricant (W m-r K-l) thermal inductive of roller a and b (W m-l K-r) variable 4SU(~/8~+’ nodes consistency of the fluid (N s m-‘) consistency at operating temperature and ambient pressure dimensionless consistency, 6 =mlmo power law exponent number of nodes used in the linear system pressure (Pa) maximum hertzian pressure (Pa) dimensionless pressure P=plp, mass flow rate per unit length (kg s-l m-‘) dimensionless mass flow rate per unit length, Q=q/p~u~R equivalent radius of contact (m) slip-to-roll ratio, SR = 2(ub -u.)& -I-utr) temperature in the film (“C) ambient temperature average entrainment rolling speed (m s-l), u~=(u.+-u~)/~ velocity of the lubricant surface velocity of roller a and b (m s-‘) a ==u/(ua +z&) dimensionless speed parameter, U=mou/[E’R” dimensionless load parameter, W= w/E’R abscissa along rolling direction (m) dimensionless abscissa, X=xib outlet meniscus distance minimum value of X in mesh coordinate across the film (m)
Greek symbols piezoviscous coefficient (m’ N-‘) thermal-viscosity coefficient of lubricant ; shear strain rate y=au/ay elastic deformation at node i (m) i dimensionless elastic deformation 4 reference viscosity rl 8 dimensionless temperature, 0 = T/T, dimensionless mean temperature of the film 4n temperature of roller a and b @d,e, V Poisson’s ratio density (kg m-‘) P ambient density of the lubricant (kg mm3) PO relative density, fi=$p,, ri relative density where H-H, PC shear stress (N m-‘) thermal expansivity of lubricant ;
Wear, 154 (1992) 77-93
Thermal elastohydrodynamic lubrication sliding contacts with a power law fluid Tsann-Rong
of rolling and
Lin
Department of Mechanical Manufacturing Engineering National Yun-Lin Instihde of Technology, Hu-Wei 63208 (Taiwan) (Received
April 24, 1991; accepted
June 7, 1991)
Abstract
A modified Reynolds equation for compressible fluids which obey the power law model was derived. Full numerical solutions were obtained by coupling the equation for lubrication with those for film shape and energy between two surfaces with simultaneous rolling and sliding motion. The effects of various power law indexes and slide-roll ratios on lubricant temperature, pressure, film thickness and friction coefficient were investigated. The results indicate that temperature effects are significant and cannot be neglected.
1. Introduction
Common lubricants exhibiting non-Newtonian behaviour include polymer-thickened oils, greases, and natural lubricating fluids which appear in animal joints. These lubricants violate the Newtonian postulate which assumes a linear relationship between shear stress and rate of shear. In recent years, a modified Reynolds equation for a power law fluid [l-5] has been derived. The power law model has received attention because of its simplicity and potential for describing many lubricants such as silicone fluids and polymer solution. The power law index n is essential in the study of power law fluid flow problems. For 12= 1, the fluid is Newtonian; for n < 1, the fluid exhibits pseudoplastic behaviour; for n > 1, the fluid is a dilatant. Dien and Elrod [l] examined non-Newtonian lubrication, and developed a perturbation expansion for velocity and pressure fields under Couette-dominated flow. Sinha and coworkers [2-4] analysed the isothermal non-Newtonian lubrication problem theoretically with rigid rolling disks under pure rolling conditions. Wanget al. [5] analysed the isothermal elastohydrodynamic lubrication problem using a power law model under pure rolling conditions. In these papers [2-51, for a power law model fluid, there are two regions (d&lx>0 and dpl dx < 0) with different Reynolds equations. The viscosity is assumed to vary exponentially with pressure. The solutions were limited to pure rolling conditions and incompressible fluids. For most practical lubricants, the viscosity and density depend strongly on temperature. For this reason, thermal effects must be considered, for they affect the film thickness and all other performance characteristics. Cheng and Sternlicht [6] developed a numerical scheme to handle the thermal Newtonian problems of elastohydrodynamic lubrication in rolling and sliding contacts. Ghosh and Hamrock [7] obtained a numerical solution for thermal Newtonian EHL of line contacts using the finite difference formulation. They analysed the film shape by subjecting each rectangular area to uniform pressure.
0043-X48/92/$5.00
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78
Recently, Najji et al. [S] presented a generalized Reynolds equation for nonNewtonian thermal hydrodynamic lubrication. Yang and Wen [9] obtained a modified generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic or hydrodynamic problems, whether the film is in a steady or a transient state. Although the previous generalized Reynolds equations can incorporate most rheological laws found in the literature, they cannot include the power law model for rolling and sliding contact. In this paper, a thermal non-Newtonian fluid analysis including effects of pressure, film thickness, rise in temperature and coefficient of sliding friction in rolling and sliding contact, is presented. The isothermal non-Newtonian results obtained by Houpert and Hamrock [lo] were modified slightly to suit the power law model which is derived by a perturbation method [l], and the proper energy equation for thermal situations is considered. A mean temperature across the film in each section is calculated by an approximate method. Then the lubrication equation is coupled with equations for film shape and energy, and they are solved simultaneously by a numerical method. 2. Mathematical
analysis
In a lubricating system, the primary objective is to develop a lubricating oil film to support a moving body with a load on it. A thin film in an EHL problem is under such a high pressure that the density and viscosity would create substantial variations in a real application. The pressure is so high on a very narrow film that even liquid oil is characterized as a compressible fluid. In this study the numerical scheme developed by Houpert and Hamrock [lo] was revised to suit the features of power law fluids. The derivations and the criteria for computation are based on the following assumptions: (1) elastic deformation of the surface is calculated for a semi-infinite soiid under plane-strain ~nditions; (2) oil side-leakage is neglected owing to the relatively broad width in this direction; (3) the boundary conditions needed to depict the narrow region are P=O at a large distance from the high pressure zone, and P=aPbX=O at the cavity outlet. 2.1. Flow equation From an order of magnitude analysis, we obtain the Navier-Stokes equation which is consistent with the basic assumptions of creeping flow. These equations and their boundary conditions are given as (see Appendix A)
ar
ap -=a~
ay at y=O, u=u, at y-h, u=ub
In the one-dimensional power law model is n-1
au
II
r=rnay
au
ay=qay
case, the constitutive
equation
for the fluid satisfying the
*au
where q*=rn/au/ay~-' is usually called the equivalent viscosity of the fluid. m is the consistency coefficient and n is the power law exponent. The fluid will either have zero or infinite viscosity outside the contact for values of II other than unity. A modification of the power law model is proposed [ll]. At a low shear rate (y-0) the fluid is expected to behave as a Newtonian fluid. At a high shear rate (+-, a)
79
the lubricant becomes Newtonian again with the same viscosity as the base lubricant. These two cases are not considered here. Substituting eqn. (2) into eqn. (l), we obtain aP a -=-
ax
ay
au
( 1
(3)
‘7ay
The asymptotic expansion is introduced to denote all related properties in terms of the small dimensionless parameter E. The x-component velocity is therefore given in the form u=u*+cu~+...
(4)
The velocity gradient
is then approximated
as the asymptotic
form
or we can write
The apparent series
viscosity and fluid-film pressure
are expressed in the forms of asymptotic
p=po+q1+... Assuming
(6) Couette-dominated
flow, the pressure
gradient
can be written
as [l]
-ap =e- a+
ax
ax
By substituting eqns. (4) and (6) into eqn. (3) and by equating power of E, one obtains
a
* ( )
G q”
au0 -= ay
coefficients
of equal
0
(8) The solution for eqn. (7) associated y = h, u =ub, gives the velocity u. as (*b
-
uo= -y-t*, h
analysis of eqn. (8) gives a solution 1
kn[(&,-&)/hr-’
y-0,
u=u.
and
1
for the first-order
velocity u1
3 cv’ -yh) aX
Thus, the velocity with the limiting
* =
conditions
*a)
An analogous *I=
with the boundary
2
v_yh)+
%n[(ub - u,)lhr- l 3.X
condition b$dy+,,
that E is sufficiently
small is written
as
(11)
80
By substituting the velocity function into the continuity equation and assuming ph = p,h, where p = ap/&x= 0, the modified Reynolds equation for a power law fluid is given as
hn+2 ap _-+!i?$%)“(h_!!$=)
SRI-”
(12)
12mn ax
Equation n = 1. The (12). The lubricants,
(12) can be reduced to the Newtonian-based Reynolds equation by putting result for pure rolling conditions from ref. 5 cannot be obtained from eqn. slide-to-roll ratio, which is the parameter which includes non-Newtonian is defined as a function of the relative tangential velocities of two surfaces:
SR = 2(ub - u,)/(u, + z+,) 2.2. Film thickness The elastic deformation
6 at any point x on the surface
is expressed
as
XC,“d &=-&
p ln(x-x’)’ s xmin
or in dimensionless
dx’
(13)
form as
Xcnd &-L
P ln(X-X’)2dX’ 2lr s xlni.
- $ In
(14)
A more accurate form of dimensionless elastic deformation & at node i was derived by Houpert and Hamrock [lo] as a function of the pressure Pi and the influence coefficient D, $=
$Dijpi-a
In
(15)
j-l
The film shape between parabolic approximation. written as Hi=Ho+
7
two elastic bodies The dimensionless
in line contact film thickness
may be expressed by a at each node i is then
(16)
+ 2Di,P, j-l
2.3. Energy equation The temperature distribution within the lubricant film is determined using the energy equation with appropriate boundary conditions. The energy equation for linecontact problems, neglecting heat convection across the film and conduction along the film, can be expressed as (17) By introducing written as
the dimensionless
variable
to eqn. (17), the energy
equation
can be
n+l
(18)
81
where m&4, + u$ kT
Q2=
f
Q3=
II-1
+’
R
0 7;z
0
@.&! (U +%,) kfR2
=
Integrating eqn. (18) in they direction twice, we obtain a two-dimensional temperature distribution which varies along both thex and y directions. As for Cheng and Sternlicht’s first approximation [6], we take the mean values of consistency m, density of lubricant p and temperature 0 along they direction. A mean temperature across the film thickness is introduced to replace the local temperature and is defined by 1 e,,.,(x)=
6’
s
dY
0
The average form of the energy equation written as
1
+ : ‘@H’-”
in terms of the mean temperature
is thus
1 36(1-~HJ$J)2
(n+3)(n+2)
x{I[3(1g+ q3+ I[? -3(,- !!&)]I”“) 1 -QfiH’-“(n+3;(n+2) Xii[,(lEquation
e
=
$$)(ZY-I)+
?]I””
_
?.-Ql~2?5
+
(19) dPldX=0.
It can be shown as follows
&Q-l-n
(20)
2
From ref. 12, the non-dimensional given as
em= 1.0
dY
(19) is not valid for PH=&H,where
h+eb
m
36(1-r.5~H,/~2
atX=-co
boundary
conditions
for the energy equation
are
2.4. Sliding friction coefficient From eqn. (3) the shear stress is
and the friction
coefficient
due to the sliding per unit width is given by
i_Jm(y)^dir/w Also, by introducing
(22)
the non-dimensional
parameters,
f can be expressed as (23)
2.5. Numerical analysis for the Reynolds equation If the side-leakage effect is neglected, owing to the extremely small high pressure contact region, the dimensionless Reynolds equation for a power law fluid of rolling and sliding contacts can be written as
with u=
fi=
m0uF -
K=48U(~l8v+’
E’R”
!E
E
p’
m0
PO
and the thin film thickness Hi=H*+
u,+ub u=s 2
x: 2
in dimensionless
form gives
N + EDijPj j-l
It is well recognized that the consistency is dependent on fluid temperature and pressure [13]. A commonly accepted relationship which describes the viscosity-temperature-pressure dependence is the Barus equation (25) The density
pi in the thermal
fluid film is related
to the local pressure
and mean
83
temperature.
The proper
o.6x10-9Hpi jYii= l-!? 1 + 1.7 X lo-‘&Pi [
expression
at X=X,,
P=aP/tlX=O
at X=X,,
(26)
{l.O-$!cT,(Bd--1.0)) I
The appropriate boundary conditions, on the cavitation boundary, are P=O
is given as
including
the Reynolds
condition
generally
used
The Reynolds equation is solved by iteration using the Newton-Raphson algorithm. The unknowns at present include the Iocation of the outlet meniscus X, the number of nodes for which the film pressure exists N, the film thickness constant Ho, the value of cS,E&where dPldx=O, and the node pressure Pi O-2, 3, _.. N). If the superscripts n and o respectively denote the new and old values corresponding to two successive iterations, the unknowns can attain convergent solutions by the relations (6%&) Pin
= =
WW + [Wk.%.)1 z-y+ (APj)” j=2,
ffOn
=
HJ0+(AH,)"
The Newton-Raphson j (i=l, 2, . . . IV):
algorithm
3, . . . . N
(27)
results in a set of N algebraic
equations
for the nodes
(28)
vi -saq
n+2 )7
a[tdPldx)il
_K
- fii
_
arii,
H._
apj (
3Pj
Dij- Ache [
I
q/ii) apj )
Since an equivalent number of equations additional load condition is needed:
jgq(&Y =(AV
is required
to solve N+ 1 unknowns,
an
(2%
where the weighting factor Cj is derived from the inte~ation of pressure at each subregion [lo]. The pressure adopted in this study is described as a linear function. Thus,
84
a linear system of N+ 1 equations, for the solution of N-I-~ unknowns, is arranged in a compact matrix. All unknowns can converge to solutions if the correct initial values are used.
3. Computational
procedure
The arrangement needed to solve the thermal non-Newtonian elastohydrod~amic lubrication problems requires the isothermal cases to be solved first. The isothermal results are thus used as the initial conditions for further thermal consideration. The oil temperature is given by the mean temperature across the film. The temperature gradient in eqn. (19) is expressed in a forward difference form to be consistent with the isothermal case where the pressure gradient is also denoted by a forward scheme. The last term is integrated using Simpson’s integral method. Equation (21) can be integrated using the trapezoidal rule. The singularity at X=X’ is removed by assuming a linear function for the conduction term of the integration equation (21). Coupled solutions were considered to be convergent when the combined solution conserved flow from the inlet to the exit region in addition to satisfying the convergence criteria associated with various equations. The iterative method combined with an underrelaxation factor (the optimal value is 0.8495) can accelerate convergence for the temperature field. The iterative procedure is stopped to give the final temperature distribution when the errors in computing the temperature in the next iterative procedure become less than 0.0005. It takes 30-40 iterations to reach the convergent solution.
4. Results
and discussion
A modified Reynolds equation was derived which is based on a power law fluid with rolling and sliding contacts. This modified Reynolds equation is reduced to the Newtonian-based Reynolds equation if the power law index n is equal to unity. Table 1 and Figs. l-11 give the results for the power law model from which we can discuss the effects of the power law index n and sliding ratio SR on thermal EHL. The point of cavitation and minimum film thickness are given in Table 1 for various values of n and SR. For the compressible fluids, the point of cavitation Xend moves away from the centre-line of the contact as it increases. Prasad et al. [13] solved problems of compressible fluid film lubrication of rollers by power law lubricants. The tendency of the cavitation point to move, which is affected by the vaiue of n, is the same as that found by the author. Former researchers [Z-5] solved incompressible fluid problems in which the cavitation point moves towards the centre of the contact as n increases. The performance of a lubricating system with load W= 1.64 x 10e5, material parameter G = 3500 and velocity U= 1.0 X 10ml’, is fixed a$ constant values. The temperature increases as the power law index n increases for any specified rolling and sliding speed; this can be seen from Fig. 1. This result can be interpreted physically. An increase in n signifies an enhanced effective viscosity. Thus the resistance to motion would increase, leading to a higher viscous dissipation. The case of pure rolling conditions is not considered here, because the temperature rise is small in pure rolling cases, and the maximum temperature rise for the mid-film temperature occurs on the far left at the beginning of the inlet zone [7]. The variation of pressure under thermal non-Newtonian conditions is shown in Fig. 2. As the power law index n increases, the pressure increases, and the pressure spike moves towards the centre of contact.
85 TABLE
1
Numerical
results
n
SR
X end
h&R
0.9 1.0 1.1 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.0 1.0 1.0 0.9 0.9 0.9
1.2 1.2 1.2 1.2 0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0 0.5 1.0 1.5 0.5 1.0 1.5
1.10199 1.20611 1.29878 1.41375 1.37016 1.38617 1.39461 1.40722 1.26619 1.27850 1.28318 1.29452 1.19107 1.20413 1.20951 1.10116 1.10169 1.10235
7.434X lo+ 17.817x 1O-6 32.927 x 1o-6 59.141 x 10-6 54.539x10-6 56.530x lo+ 57.782 x lo+ 58.599x lo+ 32.118x1O-6 32.639x lO+ 32.878 x lo+ 32.958 x lo+ 18.513 x lo+ 18.046X lo+ 16.958 x lo+ 8.187x 1O-6 7.613 x 1O-6 7.163 x 1O-6
-4
-2
0
1.044 x lo-’ 2.469 x 1O-5 4.456 x lO+ 7.747 x 10-S 6.990x10-’ 7.279 x 10-T 7.480 x 10-s 7.638 x lo-’ 4.212 x lo-’ 4.307 x 10-5 4.372x lo-’ 4.420 x lo-’ 2.483 x 1O-5 2.474 x lo-’ 2.460 x lo-’ 1.117x 10-S 1.059 x 10-S 1.026x 1O-5
0.178 0.427 0.789 1.417 1.307 1.354 1.384 1.404 0.770 0.782 0.788 0.790 0.444 0.432 0.406 0.196 0.182 0.172
0.026 0.037 0.049 0.058 0.035 0.046 0.053 0.056 0.026 0.036 0.042 0.047 0.022 0.034 0.039 0.015 0.024 0.028
2
X-coordinate
Fig. 1. Temperature profile vs. different power law indexes from thermal non-Newtonian analysis. Non-dimensional parameters are W= 1.64X 10v5, G=3500, U= 1.0X lo-**, SR- 1.2: curve 1, n = 0.9;curve 2, n = 1.0, curve 3, n = 1.1; curve 4, n = 1.2.
86
i
5
0.0
$ 0.7 ce 0.6
-4
-2
0
2
X-roordinate
Fig. 2. Pressure profile W. different power law indexes from thermal non-Newtonian analysis. Non-dimensional parameters are W= 1.64X lo-‘, G=35OO, U= 1.0X lo-“, SR= 1.2: curve 1, n =0.9; curve 2, n = 1.0; curve 3, n = 1.1; curve 4, n = 1.2.
Then we find that the location
of the pressure
spike is the same as that of the
temperature spike because viscosities and densities are taken to be functions of pressure and temperature. The minimum film thickness becomes thicker as the power law index n is increased, which can be seen in Fig. 3. Figures 4-6 show that the higher the slide-to-roll ratio SR, the higher the temperature distribution for the three cases n = 1.2, n = 1.0 and n =0.9. The generation of heat in the sliding case due to the shear strain rate is greatly increased in the contact region, but is not significant in the inlet region. Owing to the effect of shear strain rate on film thickness, the lub~cating system of the non-Newtonian lubricants causes either thinner or thicker fihns. A power law lubricant results in both shear thinning and shear thickening effects. When n> 1, as the slide-to-roll ratio SR rises, the equivalent viscosity increases. This is the shear thickening effect. When nor; 1, as the slide-to-roll ratio SR increases, the equivalent viscosity decreases. This is the shear thinning effect. As the slide-to-roll ratio SR increases, the rnin~~ film thickness becomes thinner when nzzl, and conversely becomes thicker when n > 1, as shown in Figs. 7-9. When n 2 1, the behaviour of the minimum film thickness is the same as that of viscoelastic-plastic non-Newtonian fluids. The amount of oil flow increases relatively with the increase in power law index n. This phenomenon is explained by the increase in film thickness at a high power law index n. Figure 10 shows the mass flow rates for different power law indexes IZ. From Table 1 we see that the mass flow rate decreases as the slide-to-roll ratio SR increases when IZ5 1, but it increases when n > 1. The sliding friction coefficient values at different slide-to-roll ratios obtained by isothermal non-Newtonian analysis (dashed line) and thermal non-Newtonian analysis (solid line) are shown in Fig. 11. According to the thermal non-Newtonian analysis, the friction coefficients decrease with increasing
X-coordinate
-6.4
0.4
Od
Ii?
1.6
-4
-2 X-roordi
nate
parameters
-6.6
Fig. 4. Temperature profile 21s.different slide-ta-roll ratios from thermal non-Newtonian anaIysis. Non-dimensional G=3500, U=1.0~10-‘~, n=12 curve 1, SR=O& curve 2, SR=0.9; curve 3, SR=1.2.
-1.2
parameters
-Id
6.2 0.t
1.2 1.1 I 0.9 0.6 0.7 0.6 0.5 0.4 6.2
lb 1.7 1.6 1s 1.4 I.3
Fig. 3. Film thickness profile ~8. different power law indexes from thermal non-Newtonian analysis. Non-dimensional G=3500, U=l.OXlO-“, SR=1.2: curve 1, n=O.9; curve 2, n=l.O; curve 3, n=l,l; curve 4, n=1.2,
E *G
*?. 5
z v: 8 ”
it1 2 1.9
u
13
2
are W= 1.64 X IO-‘,
are WY=1.64 X 10e5,
0
-0.5 i.5
45
25
X-coordinate
-1.5
-0.5
parameters
0.5
Fig. 6. Temperature profile ~1s.different slide-to-roll ratios from thermal non-Newtonian analysis. Non-dimensional G=3500, U= 1.0x lo-“, n =0.9: curve I, SR =0.5; curve 2, SR= 1.0; curve 3, SR= 1.5.
X-ccmrd i na t.r
-1.3
parameters
-2.5
Fig, 5. Temperature profile vs. different slide-to-roll ratios from thermal Newtonian analysis. Non-dimensional G=3500, U=l.OXIU-ll, n=l.O: curve 1, SR=0,8; curve 2, SR=1.2; curve 3, SR=l.S+
-a5
t.5
are W= 1.64 X
10e5,
are W= 1.64 X 10--5,
0.5
,
,
,
,
,
,
0
,
X-coordinate
-6.4
, 0.4
,
, 6.6
,
, 12
, 1.6 -12
-0.6
-6.4
X-coordinate
0
parameters
-0.6
Fig. 8. Film thickness profile ‘us. different slide-to-roll ratios from thermal Newtonian analysis. Non-dimensional G=3500, U-1.0X10-“, n=l.O: curve 1, SR=O.B; curve 2, SR=1.2; curve 3, SR=1.5.
,
DA2-
0.44-
0.46-
0.46-
0.1-
0.52-
0.54-
0.56-
parameters
-12
E .+ Lr
0.56-
0,6-
O&z-
0.64-
0.66-
0.66-
0.7-
0.72-
0.747
ratios from thermal non-Newtonian analysis. Non-dimensional curve 2, SR=0.9; curve 3, SR=1.2.
,
I
Fig. 7. Film thickness profiIe vs. different slide-to-roll G=3500, U=l.OXlO-“, n=1.2: curve 1, SR=0.6;
-1.6
1.3:
1.4-
1.9-
1.6-
1.7-
1.6-
1.9-
2-
L1 -
22-
2.2 -
0.6
12
are W=L64X
10a5,
are W= 1.64X 10m5,
0.4
90
c
91 0.16 a17 0.16 2 c t
0.15 0.14
1;
0.13
2
0.12
5 E
0.11 0.1
;,
0.69
.F
a06
r=
0.67
b! 2 .L
0.06
2
0.66
a# 0.03 a02 aof 0 0
6.2
a4
a6
slide-roll
a6
1
ratio
SR
1.2
i.4
Fig. 11. Sliding friction coefficient for different power law indexes: curves 1, n=0.9; curves 2, n = 1.0, curves 3, n=l.l; curves 4, n= 1.2. The dashed lies show the results of isothermal analysis and the solid lines show the results of thermal analysis.
slide-to-roll ratio. Also, the friction coefficients remain constant under a high slideto-roll ratio SR for all power law indexes n. The low shear rate viscosity at high temperatures is the main cause of the decrease in friction coefficient.
5. Conchsions A numerical solution for the problem of thermal non-Newtonian elastohydrodynamic lubrication of rolling and sliding contacts has been developed. It calls for simultaneous solution of the thermal modified Reynolds equation, the elasticity equation, the energy equation, and the transient heat conduction equation in the solids. A compressible power law fluid is used to give more extensive and more believable results. The consistency and density are assumed to vary with temperature and pressure. The normal perturbation method is used to derive lubrication equations for the nonNewtonian power law model. By reviewing the procedures described in the analysis, thermal non-Newtonian power law model EHL problems are dominated by the power law index n and slide-to-roll ratio SR. The results obtained from this paper are as follows. (1) For compressible fluids, the location of the cavitation point moves away from the centre-line of contact as II increases. (2) As the power law index IZ increases, the temperature peak and pressure peak increase, and they move towards the centre of contact. (3) As the slide-to-roll ratio SR increases, the temperature increases and the minimum film thickness decreases when n IS:1, and conversely increases when n > 1.
92 (4) As the slide-to-roll ratio SR increases, the mass flow rate decreases when n < 1, but increases when it > 1. (5) Thermal effects reduce the friction coefficient as the slide-to-roll ratio is increased for all power law indexes n.
References
6 7 8 9 10 11 12 13
I. K. Dien and H. G. Elrod, A generalized steady state Reynolds equation for non-Newtonian fluids, with application to journal bearing, ASME, J. Lubr. Technol., IO5 (1983) 385-390. P. Sinha and C. Singh, Lubrication of a cylinder on a plane with a non-Newtonian fluid considering cavitation, Trans. ASME J. Lubr. Technol., 104 (1982) X8-172. P. Sinha and A. Raj, Exponential viscosity variation in the non-Newtonian lubrication of rollers considering cavitation, Weur, 87 (1983) 29-38. P. Sinha, J. B. Shukla, K. R. Prasad and C. Singh, Non-Newtonian power law lubrication of lightly loaded cylinders with normal and rolling motion, Wear, 89 (3) (1983) 313-322. S. H. Wang, D. Y. Hua and H. H. Zhang, A full numerical EHL solution for line contacts under pure rolling condition with a non-Newtonian rheological model, ASME J. Tribal., 110 (1988) 583-586. H. S. Cheng and B. Sternlicht, A numerical solution for the pressure, temperature, and film thickness between two infinitely long, lubricated rolling and sliding cylinders, under heavy loads, J. Basic Eng., (1965) 695-707. M. K. Ghosh and B. J. Hamrock, Thermal elastohydrodynamic lubrication of line contacts, ASLE Trans., 28 (2) (1985) 159-171. B. Najji, B. Bou-Said and D. Berthe, New formulation for lubrication with non-Newtonian fluids, ASME J. TtiL, III (1989) D-34. P. Yang and S. Wen, A generalized Reynolds equation for non-Newtonian thermal elastohydrod~~ic lubrication, ASME J. T&w& 112 (1990) 631-636. L. G. Houpert and B. J. Hamrock, Fast approach for calculating film thickness and pressures in elastohydrodynamically lubricated contacts at high loads, ASME J. TriboI, 108 (1986) 41 l-420. 0. Isaksson, Numerical analysis of elastohydrodynamic contacts using power-law lubricant with special reference to water-based hydraulic fluids, ASLE Trans., 30 (4) (1987) 501-507. J. C. Jaeger, Moving sources of heat and temperature at sliding contacts, J. Proc. R. Sot. N.S. W, 76 (1942) 283-224. D. Prasad, P. Singh and P. Sinha, Non-uniform temperature in non-Newtonian compressor fluid film lubrication of rollers, ASME J. Tribal., 110 (1988) 653658.
Appendix A: Nomencia~~ b c c,,
ci Dij
cb
half hertzian length (m), b=R(8Ww)‘R specific heat of lubricant (J kg-’ K-‘) specific heat of roller a and b (J kg-’ K-‘) weighting factors used to integrate P influence coefficient used to calculate elastic deformation at node i due to pi
E’
equivalent Young’s modulus (Pa), f
=
l--y;2 + Eb
G h H
materials parameter, G = CYE’ film thickness (m) dimensionle~ film thickness, H = hlctlb’
93
Ht? HO k kmkb K
4i m MO
rii n
N P pd” 4
Q
R SR T To US u u,,
@b
22 u
W x X X end -Knin Y
dimensionless film thickness where dP/dX=O dimensionless constant used in calculation of H thermal conductivity of lubricant (W m-r K-l) thermal inductive of roller a and b (W m-l K-r) variable 4SU(~/8~+’ nodes consistency of the fluid (N s m-‘) consistency at operating temperature and ambient pressure dimensionless consistency, 6 =mlmo power law exponent number of nodes used in the linear system pressure (Pa) maximum hertzian pressure (Pa) dimensionless pressure P=plp, mass flow rate per unit length (kg s-l m-‘) dimensionless mass flow rate per unit length, Q=q/p~u~R equivalent radius of contact (m) slip-to-roll ratio, SR = 2(ub -u.)& -I-utr) temperature in the film (“C) ambient temperature average entrainment rolling speed (m s-l), u~=(u.+-u~)/~ velocity of the lubricant surface velocity of roller a and b (m s-‘) a ==u/(ua +z&) dimensionless speed parameter, U=mou/[E’R” dimensionless load parameter, W= w/E’R abscissa along rolling direction (m) dimensionless abscissa, X=xib outlet meniscus distance minimum value of X in mesh coordinate across the film (m)
Greek symbols piezoviscous coefficient (m’ N-‘) thermal-viscosity coefficient of lubricant ; shear strain rate y=au/ay elastic deformation at node i (m) i dimensionless elastic deformation 4 reference viscosity rl 8 dimensionless temperature, 0 = T/T, dimensionless mean temperature of the film 4n temperature of roller a and b @d,e, V Poisson’s ratio density (kg m-‘) P ambient density of the lubricant (kg mm3) PO relative density, fi=$p,, ri relative density where H-H, PC shear stress (N m-‘) thermal expansivity of lubricant ;