thermal analysis of · for the ranges of parameters usually found in practice. by using a...

Shao Wang Research Assistant,

Department of Mechanical and Industrial Engineering.

C. Ousano Professor of Mechanical Engineering,

Department of Mechanical and Industrial Engineering. Mem. ASME

T. F. Conry Professor of General Engineering

and Mechanical Engineering, Department of General Engineering.

Fellow ASME

University of Illinois at Urbana-Champaign, Urbana, III. 61801

Thermal Analysis of Elastohydrodynamic Lubrication of Line Contacts Using the Ree-Eyring Fluid Model A thermal Reynolds-Eyring equation is derived for elastohydrodynamic lubrication of line contacts. A control volume approach is used to analyze the inlet region where back-flow occurs. Numerical results are obtained and used to develop a formula for the thermal and non-Newtonian (Ree-Eyring) film thickness reduction factor. Results for maximum temperatures and traction coefficients are also presented. The pressure dependence of lubricant thermal conductivity is found to significantly affect the maximum lubricant temperature.

Introduction The strong dependence of lubricant viscosity on temperature

significantly influences the traction and film thickness between surfaces in rolling and sliding contacts in the elastohydrodynamic (EHD) lubrication regime. Under high speeds and/or loads, the temperatures in the lubricant can reach critical values, causing a lubricant breakdown and subsequent surface failure. For over three decades, determining the operating conditions that correspond to the onset of these failures has been one of the most important goals in the area of tribology.

The first full numerical solution for obtaining film thickness data in line contacts was obtained by Dowson and Higginson (1959). Their results are in general agreement with the experimental data of Crook (1957, 1958) and Sibley and Orcutt (1961). The effects of heat generation and resulting temperature rise in the (Newtonian) lubricant film were included in the analyses of Cheng and Sternlicht (1965) and Wilson and Sheu (1983). As expected, these analyses predicted film thickness values lower than those predicted by isothermal solutions. In these thermal analyses, the film thicknesses predicted by isothermal models are modified by a thermal reduction factor. Design practice today frequently uses this approach to predict film thickness in EHD contacts.

Because the analyses discussed above are all based on a Newtonian fluid model, it has been found that a reasonable prediction for fluid traction cannot be obtained. The experimental work of Johnson and Cameron (1967-1968) showed that, with an increase in the slide/roll ratio, the traction increases to a maximum and then decreases. Crook (1961) at-

Contributed by the Tribology Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS and presented at the Joint ASME/STLE Tribology Conference, Toronto, Canada, October 7-10. Manuscript received by the Tribology Division February 26, 1990; revised manuscript received June 1, 1990. Paper No. 90-Trib-27.

tempted to explain this phenomenon by considering the effects of shear heating on the viscosity of a Newtonian fluid. However, this theory agrees with experiments only when the pressure is relatively low.

To resolve the discrepancies between the theories and experiments for the traction behavior, a number of models have been proposed. One of the accepted models is that proposed by Johnson and Tevaarwerk (1977), a nonlinear Maxwell model of the form

1 dr T0 . y = GJt + ^Smh ©• (1)

Johnson and Tevaarwerk also showed, through their experimental data, that the elastic term (first term) in equation (1) could be neglected when the shear strain rate, 7, is reasonably large. In this case, equation (1) is simplified into the form,

7 = — sinh V ©• (2)

This non-Newtonian rheological equation is based on a Ree-Eyring model (Ree and Eyring, 1955, and Hahn et al., 1958). It characterizes a shear-thinning behavior in the sense that the slope, dr/dy, decreases as 7 increases. Also, as T0-*OO, the non-Newtonian behavior vanishes and equation (2) approaches the Newton's law of viscosity. The traction tests conducted at controlled surface temperatures by Conry et al. (1980) confirmed the validity of equation (2) for relatively high shear strain rates.

The non-Newtonian behavior of a lubricating fluid not only reduces traction under conditions of high shear strain rates, but also reduces film thickness compared to those predicted by Newtonian models. By employing equation (2), Conry et al. (1987) obtained an isothermal Reynolds-Eyring equation. Their numerical solutions showed an appreciable reduction of

2 3 2 / V o l . 113, APRIL 1991 Transactions of the ASME

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Fig. 1 Description of the contact region

the film thickness due to the non-Newtonian effect under combined rolling and sliding conditions. These results are in good agreement with those of Gecim and Winer (1980), who used an Ertel approximation (Ertel, 1984 and Grubin and Vino-gradova, 1949) and a shear-thinning model to analyze non-Newtonian effects.

The film thickness reduction at high shear strain rates is due to the combination of thermal and non-Newtonian effects. These combined effects were studied numerically by Wang and Zhang (1987) with the assumptions that the lubricant viscosity is constant across the film and that the lubricant obeys a limiting shear rheological model similar to that proposed by Bair and Winer (1979).

The temperature results for EHD contacts obtained numerically by previous investigators, such as those by Cheng and Sternlicht (1965) and those by Conry (1981), for a non-Newtonian fluid with assumed film shape, are based on the assumption of a constant thermal conductivity of the lubricant. However, Richmond et al. (1984) found experimentally that the thermal conductivity of a mineral oil increased to twice its ambient value when pressure was increased to 1 GPa. This finding indicates that the actual temperature rise in the contact may be much less than that calculated with a constant thermal conductivity.

In this paper, a thermal Reynolds-Eyring equation is derived by utilizing the non-Newtonian constitutive law, equation (2), and considering the effects of the variation of viscosity with temperature across the film. Numerical solutions are obtained for the ranges of parameters usually found in practice. By using a regression analysis, a formula for the thermal and non-Newtonian (Ree-Eyring) film thickness reduction factor, Cn

is obtained. The change of the lubricant thermal conductivity with pressure is also included in the analysis.

Governing Equations

Thermal Reynolds-Eyring Equation. In deriving a Reynolds-Eyring equation, Conry et al. (1987) assumed a constant lubricant viscosity across the film. In order to consider the effect of temperature variation across the film, and resulting variations of viscosity, a thermal Reynolds-Eyring equation is derived.

Referring to Fig. 1 and considering the equilibrium of a fluid element, the momentum equation can be written as

dT„/dz=dp/dx. (3)

Integration of equation (3), assuming that p=p(x), gives Txz = Ti+ zdp/dx. (4)

Under the assumption that 8w/dx«du/dz, equation (2) is rewritten as

du r0 — = — sinh az 1} & ) •

(5)

Substituting equation (4) into equation (5) and integrating, the expression for u(x, z) is written in terms of a dimensionless function, A, and a midlayer shear stress, T„„ as

u(x, z)=Ul + [z h dp 1 I sinh J° 2-q dx A I To h

dt, (6)

Nomenclature

a = [8RfV/(%E')]m, half Hertz contact width, m

Cr = H0/HOJN, thermal and non-Newtonian (Ree-Eyring) film thickness reduction factor

Cfp = constant-pressure specific heat of the lubricating fluid, J/kg-K

cu c2 = specific heats of solid surfaces 1 and 2, J/kg-K

Eu E2 = moduli of elasticity of solid surfaces 1 and 2, Pa

E' = 2/[(l - v\)/E{ + (1 - "l)/E2], equivalent elastic modulus, Pa

G = shear modulus of fluid, Pa gt = [a2W*7(rj0Cy?2)]1/2, dimensionless viscosity

parameter (Johnson, 1970) g2 = [a

2WE'/(2vR)]W2 = ap0, normalized Hertz stress (Johnson, 1970)

g5 = T0a2E', normalized Eyring stress

h = film thickness, m h0 = central film thickness, m

hmia = minimum film thickness, m hojN = isothermal and Newtonian central film

thickness, m H0 = hoW/irioUoR), dimensionless central film

thickness hojfjW/irioUoR), dimensionless isothermal and Newtonian central film thickness

Ho,iN -

•f̂ min = hja^W/iTjoUoR), dimensionless minimum film thickness

hh hb, hu = thicknesses of the lower, back-flow, and upper regions in the inlet, m

kf = thermal conductivity of the lubricating fluid, W/m-K

k0 = thermal conductivity of the lubricating fluid at ambient pressure, W/m-K

ku k2 = thermal conductivities of solid surfaces 1 and 2, W/m-K

Z,] = r)0pul/k0, dimensionless thermal loading parameter;

L2 = fi/(apCfp), dimensionless convective thermal loading parameter;

Z,31 = pCfpko/ifliCiki), lubricant-surface property ratio for surface 1

£3,2 = Pcfpko/(fi2C2k2), lubricant-surface property ratio for surface 2

L4 = ak/{ak0), first dimensionless thermal conductivity coefficient

£5 = ft/a, second dimensionless thermal conductivity coefficient;

p = pressure, Pa p0 = [WE'/(2TTR)]1/2, maximum Hertz pressure,

Pa P = p/po, dimensionless pressure

i?i, R2 = radii of curvature of surfaces 1 and 2, m R = R1R2/(Ri+R2), equivalent radius, m S0 = temperature-viscosity index in the Roelands

viscosity relation

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where

and

h dp T'" = Ti + 2dx'

A = h dp

(7)

(8) 2T0 dx

Defining the nondimensional coordinate across the film as

r=.2z/A(x)-l , (9)

Equation (6) can be rewritten as

h2 dp 1 f f 1 u(x,&=Ul+— — - \ - [smh(Tm/T0)coshA*

4 dx A •> ~' TJ

+ cosh(T„,/T0)sinhA*-]cf^r. (10)

In this derivation, T0 is assumed to be constant across the film. The factor, 1/ij, in the integral of equation (10) is approx

imated by a quadratic function, based upon the lower-surface, midlayer and upper-surface temperatures, Tlt T„„ and T2.

l/ij(x, f)= [A? + Bt+ l]/r,m(x), (11)

where i]m(x) is the viscosity at the midlayer, f = 0 (or z = h/2), calculated from Tm\ and yl and B are functions of x, defined as

1 • * - * ' ( 1 2 )

and

B

V7i W

2 " \1?2 W (13)

Substituting equation (11) into equation (10) and integrating yields

u(x, 0=£7,

h2 dp 1

^^Ar 1 ^^ (f2 sinh Af + sinhA)

- - j (fcoshAf+cosh A) + —j (sinh Af+sinh A)

+ 5sinh(rm/r0)

A3

— (fsinhAf-sinh A)

- ~ 2 (cosh Af-cosh A)

+ — sinh (rm/T0) (sinh Af+sinh A)

+ ylcosh (rm/T0)

2

(f2 cosh Af-cosh A)

2 (flsinh Af-s inh A)+ -̂3 (cosh Af-coshA) _2_ A3

1 (fcoshAf + coshA) + 5cosh (T„,/T0)

~A

+ — cosh (T,„/T0)(cosh Af-coshA)

2 (sinh Af+sinhA)

(14)

The velocity boundary condition at the upper surface, u(x, 1) = U2, is substituted into equation (14) to give

(fi0+-4fii)sinh(Tm/T0)

Nomenclature (cont.)

T = temperature, °C T0 = bulk temperature, °C T* = (3(T— T0), dimensionless temperature

Tu Tm, T2 = temperatures of surface 1, midlayer, and surface 2, °C

Th Tb, Tu = average temperatures of the lower, back-flow, and upper regions in the inlet, °C

Tbl = temperature at the interface between the back-flow and lower regions in the inlet, °C

Tbu = temperature at the interface between the back-flow and upper regions in the inlet, °C

t = (1) time; (2) integration variable Uu U2 = velocities of surfaces 1 and 2 in the x-direc-

tion, m/s U0 = ([/, + U2)/2, rolling velocity, m/s u = fluid velocity component in the x-direction,

m/s W = load per unit contact length, N/m w = fluid velocity component in the z-direction,

m/s x = coordinate (abscissa), m

xim *out = inlet and outlet boundary positions of the solution domain, m

xs = abscissa of the stagnation point, m X = x/a, dimensionless abscissa Y = thermal and non-Newtonian (Ree-Eyring) in

fluence parameter z = (1) coordinate, m; (2) pressure-viscosity in

dex in the Roelands viscosity relation

a - pressure-viscosity coefficient for near-ambient pressure and bulk temperature of the lubricant, 1/Pa

ak = first pressure-thermal conductivity coefficient, W/m-K-Pa

|8 = temperature-viscosity coefficient for near-bulk temperature and ambient pressure of the lubricant, 1/K

ft = second pressure-thermal conductivity coefficient, 1/Pa

7 = rate of shear strain of fluid, 1/s f = 2{z/h) - 1, dimensionless coordinate across

the film •q = viscosity, Pa-s;

rjo = viscosity at ambient pressure and bulk temperature, Pa-s

Uu Vm> V2 = viscosity values at surface 1, midlayer and surface 2, Pa-s

A = [h/(2T0)]dp/dx, dimensionless Poiseuille-flow stress

vu v2 = Poisson's ratios of solid surfaces 1 and 2 £ = (Ui-U2)/U0, slide/roll ratio p = density of lubricant, kg/m3

Pu Pi = densities of solid surfaces 1 and 2, kg/m3

T = shear stress in fluid, Pa T0 = Eyring stress of fluid, Pa n = lower-surface shear stress (at surface 1), Pa rm = midlayer shear stress (at z = h/2), Pa $ = dissipation function, W/m3

234 /Vo l . 113, APRIL 1991 Transactions of the ASME


where

fio = sinhA

A

and

Q,=

Qj — fl0— "2>

3(AcoshA - sinhA)

Equation (15) can be rearranged into the form

/i(x)[exp(Tm/70)]2 +/2(x)exp(Tm/T0) +Mx) = 0,

where

f1(x) = Q0+AQi+-AQ2,

f2(.x)-4(I/,-£/2)i?„

h2

and

dp/dx

B. Mx)=-Q0-AU1 + -AU2

(16)

(17)

(18)

(19)

(20)

(21)

(22)

The expression, exp(Tm /r0) , can be solved as a root of equation (19), and j m can then be obtained. There is always a real root for a physically existing fluid field.

Using standard assumptions, the mass flow rate, M(x), in dimensionless form, is written as

M(x) ph

j : . - •(*, j v r . (23)

Substituting equation (14) into equation (23) and using equation (8), one obtains

M(x) = phUl —£— ̂ [ (1 -B)U2-3yJQ3]cosh(Tm/T0) 127?m dx

+ ^-—1(4- S)Qi + fio]sinh(rm/r0),

2 Vm

(24)

(25)

where

0 3 = - [(A3 + 6A)coshA - (3A2 + 6)sinhA]/A5.

The continuity of fluid flow under a steady-state condition requires that

d

dx M(x) = 0. (26)

Substituting equation (24) into equation (26) and rearranging yields the thermal Reynolds-Eyring equation given by

(27)

where S^x) and S2(x) are called modifying functions and are defined as follows:

d dx Ll2rimdx

+ fi-

d

~dx

2 Vm

phUi

sinh {jm/T0)S2{x)

and

S , ( x ) = [(1 - 5 ) n 2 - 3 / l f i 3 ] c o s h ( r m / T o ) ,

S2(x) = (A-B)Ql + Q0-

(28)

(29)

The midlayer shear stress, TOT, in equations (27) and (28) can be found by solving equation (19). For the isothermal condition, in which the viscosity is constant, both A and B are zero, and equation (27) is reduced to the isothermal Reynolds-Eyring equation given by Conry et al. (1987). Fur thermore , for isothermal Newtonian fluids, r0 tends to infinity, and equa

tion (27) can be shown to reduce to a conventional Reynolds equation given by Dowson and Higginson (1977).

Elasticity and Load Equations. The surface profile, or film shape, can be evaluated from the following elasticity equation.

y2 2 f*out /x—s\2

* W = *° + 2«-^k^)In(—) * (30)

The load equation is given as

p(x)dx= W. (31)

Energy Equation. Assuming conduction only in the z-di-rection and convection only in the x-direction, the energy equation of the lubricating fluid takes the form (Conry, 1981),

dT d2T _ N

pCfpU ~dx= 7 a ? + ( x ' z ) ' (32)

where $(x, z) is the dissipation function, defined as the product of the shear stress and the viscous shear strain rate, i .e.,

* ( x , z) •• 70

Tm + H) dp

dx sinh

Tm + (z~h/2)dp/dx

To

(33)

Equat ion (32) is used only for the region where the velocity component , u, is non-negative throughout the cross section of the film, i.e., the region x>xs. The energy equations for the region x<xs will be discussed later.

The boundary conditions for the energy equation, for two moving surfaces, are given by

Tx(x)

and

T2(x)

•T„ + -1

Tn + -

(irp,c1C/'|A:,)1-

J (•wp2c2U2k2f

ikf

Kf

dT

dz„

dT'

' dz

ds

=°A/ x — s

z = h

ds

yjx-i

(34)

(35)

The boundary conditions, at the cross section, x = xs, are obtained from interpolation of the temperatures found by solving the energy equations for the inlet region.

Viscosity Relation. The Roelands viscosity relation (Roe-lands et al . , 1963) used in this research is given by

r/ = exp|[ lnij0

+ 9.668] H + 1.961x10' )"(£ir-H- <*>

Defining the Roe lands reference viscosity as -qR = exp ( - 9 . 6 6 8 ) = 6.328 x 10" 5 (Pa-s), the Roelands reference pressure as pR= 1.961 x 10s P a and the Roelands reference temperature as TR= — 135°C, equation (36) can be rewritten as

-so ( l + p / f t ) M l + f - i r ) - 1 7) = %exp ln0j0/»)*)

" > " ( ' + ^ )

(37)

• This form of the Roelands viscosity relation is dimensionless and gives an explicit expression for the rat io , rj/ijo- It can be shown that , whenp«pR and (T- T0)«{T0-TR), equation (37) reduces to the Barus viscosity relation,

7, = 7 / 0 e x p [ a p - ( 3 ( r - 7 ' 0 ) ] , (38)

where a and B are determined from the following relations.

a = zln(ij0/»)/j)/p/j, (39)

and

Journal of Tribology APRIL 1991, Vol. 113/235


Upper Region

Fig. 2 The streamlines of the flow field in the inlet region

(3 = SQln(r,0/VR)/(T0-TR). (40) Since the film thickness of an EHD contact is determined

primarily by the conditions in the inlet region where the pressure, p, and temperature rise, (T- T0), are both small, the parameters for this condition, a and /3, as defined by equations (39) and (40), together with r)0, are the most important parameters in the viscosity relation. Therefore, instead of using r)0, z and S0, the parameters i?0, a, and /3 will be used in this study to describe the dependence of viscosity on pressure and temperature.

Thermal Conductivity Relation. The dependence of the thermal conductivity upon pressure of a representative mineral oil, LAD 2201, was given graphically by Richmond et al. (1984). The data points on this experimental curve, for pressures up to 1.2 GPa, have been measured, and excellent fitting has been found with the following equation.

kf=ko + akp/(l+0kp), (41) wherea* = 2.14xlO-10(W/m-K-Pa),ft = 6.91xlO-10(l/Pa), and k0 = 0.124 W/m-K. These values of ak and flk are used in the current analysis.

System of Energy Equations for the Inlet Region. A stagnation point exists in the inlet region of an elastohydrodynamic (EHD) line contact. The stream-lines and the stagnation point, S, are shown in Fig. 2, where large arrows represent flow directions and dashed lines represent a locus of zero x-com-ponent of velocity, i.e., w = 0. The abscissa of the stagnation point is denoted by xs. In the region upstream of the stagnation point, x<xs, there exists a back-flow region where u is negative. The region, x<xs, is divided into three sub-regions, namely, the lower region, the back-flow region (middle region), and the upper region, as shown in Fig. 2. As a result of the back-flow, numerical instability arises in the common forward marching scheme for the temperature solution of the energy equation. A simple control-volume scheme therefore is presented to treat the inlet region.

Since the full solution to the thermal EHD lubrication problem involves a number of iteration loops, the solution for the inlet temperature is only one step in these iterations. In this step, it is assumed that the surface temperatures Tx(x) and T2(x), the pressure distribution p(x), and the film thickness profile h(x) are known. The viscosity of the lubricant, i}(x, z), can be calculated based on the pressure distribution given and the temperature distribution, T(x, z), found on the last iteration for the temperature solution. For a given viscosity, pressure and film thickness, the lubricant velocity distributions, u(x, z) and w(x, z), and the midlayer shear stress, rm(x), can be found. Therefore, the following derivation for the energy equations will be based on givenp(x), h(x), u(x, z), w(x, z), rm(x), T{(x), and T2(x).

Consider a control volume of width Ax, height h, and unit depth, as shown in Fig. 3(a). In this control volume, the lower region, back-flow region and upper regions are denoted by L,

T x x + Ax u

Fig. 3(a) The three regions in the control volume

ph u c, T -K u u fp u

pAxw, c, T. ' bu fp bu

PhbubcfpTb -

PAxwb/fpTbi>

pryy^T, - * -

kfAx(Tu-T2)/(hu /2)

-p huuuc lpTu+^<PhuuucfpTu>A x

k,Ax(Tb-Tu)/[(hbd+hu)/2J

~PVbVb + dJ ( fVbc fPTb>Ax

kfAx(TrTb)/[(hb + hi?)/2]

d

k - ^ - k ( A X n ) - T ) / ( h / 2 )

WJVJs T7777

Fig. 3(6) The rates of heat transfer for each of the regions

Fig. 3 The heat transfer rates for a control volume, hAx

B, and U, respectively. The average temperatures in the L, B, and [/regions are, respectively, designated as Tfcx), Tb(x), and Tu(x), and the heights are denoted, respectively, by h^x), hb(x), and hu(x).

The temperatures, Tbu(x), at the interface between regions B and U, and Tb/(x), at the interface between regions B and L are approximated by interpolation as follows:

and

Tbu(x) = (huTb + hbTu)/(hb + hu),

Tbl(x) = (hlTb + hbT,)/(hb + hl).

(42)

(43)

The heat transfer rates for the lower, back-flow and upper regions in the control volume are shown in Fig. 3(b). The arrows indicate the assumed positive directions of heat transfer. In the x-direction, only convection is considered while in the z-direction, both conduction and convection are considered. In the latter case, conduction occurs between regions B and L, between B and U, between L and surface 1 and between [/and surface 2; and convection occurs between regions B and L, and between B and U. Referring to Fig. 3(b), the law of conservation of energy, under steady state, is applied for the lower region, L. The resulting energy equation is given by

~dx (phlUlCfPTl} ~ PAxwblcfpTbl

-AryAx(r,-T1)/(V2) -kfAx(T,- Tb)/[(hb+h,)/2] + <f>,hiAx=0, (44)

where w, is the average x-component of velocity of the lower region, defined by

eh.

"/ 4Jo'"(*> z)dz, (45)

The z-component of velocity, wbl, is evaluated by employing continuity at the interface between regions B and L,

1 d , ^ Wbl= T (P"M> P ax (46)

236 /Vo l . 113, APRIL 1991 Transactions of the AS ME


and *, is the average dissipation function of the lower region, defined by

1 f"'

*'=T Jo * u z)dz- (47)

Substituting equation (43) into equation (44), the energy equation is obtained for the lower region, which is given by

hlPcfpu, — ' + f4(x)Tl+f5(x)Tb = h^,+ (2k/hl)Tl, (48)

where

fi(x) = pcfpwblhb/{hb + hi)+2kf/hi

+ 2k/(hb + h,) + cfpd(puihi)/dx, (49)

and

f5(x)=pcfpwblh,/(hb + h,)-2kf/(hb + hl). (50)

Similarly, the energy equation for the back-flow region, B, is found to be

hbpcfpub^ +Mx)T,+Mx)Tb+Mx)Tu = hb*b, (51)

where

f6(x) = -pcfpwblhb/{hb + hl)-2kf/(hb + h,), (52) Mx)=~ pcfpwblh,/ (hb + h,)

+ pc/pwbuhu/(hb + hu) +2k/(hb + h,) (53) + 2k/ (hb + hu)+cfpd(pubhb) /dx,

Mx)=pcfpwbuhb/(hb + hu)-2kf/(hb + hu), (54)

ub is the average x-component of velocity for the back-flow region, defined by

"6 = j - ]h/ u(x, z)dz. (55)

The z-component of velocity, wbu, is evaluated by employing continuity at the interface between regions B and U,

*%, = - — (puuhu), (56) p dx

and $(, is the average dissipation function of the back-flow region, defined by

*6 = r - L #(Jf,«)dz. (57)

The energy equation for the upper region, U, is found to be

huPcfpuu^ +f9(x)Tb+f10(x)Tu = hu<i>u+(2k/hu)T2, (58)

where

/ 9 (x) = -pcfpwbuhu/(hb + hu)-2kf/(hb + hu), (59)

/ i o W = -pcfpwbuhb/(hb + hu)+2kf/(hb + hu)

+ 2k/hu + cfpd (puuhu) /dx, (60)

«„ is the average x-component of velocity of the upper region, defined by

u»=iu\>.-»uu(x'z)dz> (61)

and $„ is the average dissipation function of the upper region, defined by

*«=^L/.„* (*>^- (62)

The set of ordinary differential equations, equations (48), (51) and (58), form a system of energy equations for the inlet region before the stagnation point. The average temperatures,

Tfa), Tb(x), and Tu(x) can be found by solving this system simultaneously subject to the boundary conditions given by T/=TU= T0 as x approaches negative infinity. In the numerical calculation, these boundary conditions for 7) and Tu are applied at the inlet boundary, which is taken as x-m = -3a for most of the solutions. The upper and lower surface temperatures in the inlet region for x<xs are assumed to be T0 in the calculation since the surface temperature rise is small in this region.

Numerical Considerations. The thermal Reynolds-Eyring equation, (27), the elasticity equation, (30), and the load equation, (31), are solved by using the Newton-Raphson iteration. The boundary conditions for the pressure are given by p = 0 at the inlet boundary andp = dp/dx = 0 at the outlet boundary. The integral for the elastic deformation, in equation (30), is replaced by a weighted sum of discrete pressure values in the contact. Once the pressure distribution and film thickness are found, the energy equations, (32), (48), (51), and (58), are solved with finite difference schemes. In the inlet region, for x<xs, an upwinding difference scheme (Spalding, 1972) is used to ensure numerical stability. The temperatures obtained in a given step are used to solve for the pressures and film thickness in the next iterative step of a direct iteration loop. An under-relaxation factor of 1/2 and a limitation for the maximum change are imposed on the temperatures in the iteration loop to improve convergence.

Results and Discussion

Numerical results are presented for different values of independent dimensionless groups. In order to obtain these groups, the dimensionless quantities, X* = aE' x/R = 4 ap^x/ a, Ho=(aE')2h0/R, as suggested by Johnson (1970), as well as P, f, T*, and T* = T/T0, are employed in nondimensionalizing the governing equations. A complete set of thirteen independent groups are found. These groups are gu g2, g5, £, Lu L2, •^3,1. -̂ 3,2. L4 and L5, as defined in the Nomenclature, and the material-geometry parameter, LR, Roelands pressure parameter, PR, and Roelands temperature parameter 6R, defined, respectively, as LR = R2pcfp/[k0-qRa1(E')6], PR = apR, and dR = (3(T0-TR). Note that LR, PR, and 6R only appear in the dimensionless form of the Roelands viscosity relation, equation (37). In this equation, the ratio, t)0/i}R, can be expressed as JJ0/ r\R = 64ir('LRLig22/(g\Ll). Values of the constants T\R, pR, and TR, are given in the section for the viscosity relation. All other dimensionless quantities mentioned above are functions of these independent groups. The parameters, gi and g2, were introduced and identified by Greenwood (1969) as a complete set of independent dimensionless groups for isothermal EHD lubrication of a line contact, with the dependent group as H0, which is related to H$ by H0 = H%g}/(4Tr2g\).

The physical significance of the dimensionless groups mentioned above is described as follows. The normalized Hertz pressure, g2 = ap0, is a measure of the viscosity increase with the maximum Hertz pressure. The group, glt called a viscosity parameter, describes the influence of the hydrodynamically generated pressure on viscosity. This parameter can be expressed as gx - 5a#max, where ^ax is the maximum pressure in the contact obtained by assuming an isoviscous fluid and rigid cylinders (Greenwood, 1969 and Johnson, 1970). The value of <7max is given by Martin (1916) as qmax = 0.20[Wi/(7)0U0R

2)]U2. When the quantity, g\/i = aqmmi, is much less than g2, it is possible for the contact to generate a hydrodynamic pressure of magnitude qmax without appreciably deforming the surfaces. In such a case, whether the viscosity variation has an important influence on the film thickness depends upon the value of gx/ 5. When g / 5 is much less than unity, the viscosity variation can be neglected.

The group, g5, called a normalized Eyring stress, describes the degree of shear-thinning behavior of lubricant. As g5 tends

Journal of Triboiogy APRIL 1991, Vol. 113 /237


' 0.9-

0.8-

«- 0.7-O

s" 0G"

^ 0.5-

Red

uctio

n P

' p

0.2-

0.1 -

0.0-

£

f-

= 0.1 <

1.R i

c*~~-

L-—-C^J

- ~ - ^ X

- g, = 240 - g, =495 - g, = 989

\

Fig. 4(a) The thermal and non-Newtonian film thickness reduction factor for ( = 0.1 and £ = 1.8

Fig. 6(a) Minimum-to-central film thickness ratio for { = 0.1

0.9-

0.8-

O 0.7-

o U 0 ' 6 _ H

c 0.5-o

=> 0.4-"O a)

01 0 3 -

0.2-

0.1 -

0.0-

8 o -

~~ ~"°-- ^ ^ « . ^ • « . Ns

\ ~~*—-^ "*" —

gi = 240 g, = 495 QT = 989

0.7-

0.6-

0.5-

0 4 -

\

9 i i - ' i u g, = 495 g, = 989

\ \ \ \ k \ \ \

\ \ \ \ \ \

Fig. 4(b) The thermal and non-Newtonian film thickness reduction factor for £ = 0.6

Fig. 6(b) Minimum-to-central film thickness ratio for $ = 0.6

Influence Parameter, Y

Fig. 5 Thermal and non-Newtonian film thickness reduction factor calculated from the empirical formula (straight line) with numerical results (data points)

to infinity, this behavior vanishes, and the lubricant becomes a Newtonian fluid.

The thermal loading parameter, Lu is a coefficient of the dissipation term when the energy equation, equation (32), is written in a nondimensional form. The convective thermal loading parameter, L2, is the reciprocal of the coefficient for the convection term (the term with 8T*/dX*) in the energy equation when equation (32) is written in a nondimensional form. When L2 is large, the influence of the heat dissipation on dT*/dX* is pronounced. The groups, Z,31 and L3 2, give the relative magnitudes of the thermal properties of the lu-

0 7 -

0.6-

0.5-

0.4-

X \ \ \ \

W \ \ \ X

V -X^ g, ^4U g, =495 g, =989

Li

Fig. 6(c) Minimum-to-central film thickness ratio for { = 1.8

bricant and solids. The groups, L4, and Ls, characterize the increase of the lubricant thermal conductivity with pressure.

It should be noted that the groups, LR, PR and 8R, characterize high-pressure and high-temperature performance of a lubricant while the coefficients, a and 0, characterize the performance at low pressure and near-bulk temperature. Since the EHD film thickness is primarily determined by the low pressure and near-bulk temperature conditions in the inlet region, these three groups do not significantly affect the film thickness. Furthermore, since the temperature rise in the inlet is small, L4 and L5 do not affect the film thickness significantly. Hence, H0 can be expressed as a function of the other eight groups.

238 /Vo l . 113, APRIL 1991 Transactions of the ASME


Fig. 7 Maximum midlayer temperature for different (.„ g/,, and £ (r0 = 90°C)

Fig. 8 Maximum surface temperature for different t „ g„ and f (7"0 = 90°C)

The major goal of this study is to obtain film thickness and temperature data in EHD line contacts using mineral oils and steel surfaces. Although the viscosity, TJ0, varies over a wide range for different mineral oils, the values of a and /3 do not vary significantly. Therefore, the a and /3 values are assumed to be constants and are based on a representative mineral oil, LVI-260, at a temperature of 90°C. For this temperature, the pressure-viscosity coefficient, a, is 2.1 x 10~8 Pa"1, which is obtained from the data of Galvin et al. (1964). The temperature-viscosity coefficient, (5, for this oil is found to be 0.0351 (\/K) based on the data given by Conry et al. (1979). The parameters, z, and S0, in the Roelands viscosity relation, can be calculated by using equations (39) and (40), respectively. Other representative properties for mineral oils are also based on LVI-260. Among these, all kept constant in this study, are p = 929 kg/m3, c/p = 2250 J/kg-K, A:0 = 0.125 W/m-K, as given by Conry (1981), and T0 = 2.9X 106 Pa, as given by Johnson and Tevaarwerk (1977). This value of T0 is a medium value among those of the lubricants tested by Johnson and Tevaar-werk. The lubricant is considered to be incompressible based on the results found by Goglia (1982). The material parameters of the solid surfaces are taken as those of steel to be £"= 2.28 x 101' Pa, P! = p2 = 8000 kg/m3, c, = c2 = 470 J/kg-K, and kl = k2 = 'i& W/m-K. The equivalent radius, R, is taken as 0.0395 m for all the results presented graphically in this paper.

Under the above assumptions, the dimensionless groups, L2, L},u £3,2. LR, and g5 are constants and are given by L2 = 0.799,

0.02-

0.01 -

0.00-

9i <-'iu Q l =495 q-i = 989

a ^ «. «.

Fig. 9(a) Traction coefficient for i; = 0.1

O) 0

0 c 0

0

0.02-

0.01 -

0.00-

9 l = 4 9 5 gn =989

.01 .1 1 10

Li

Fig. 9(b) Traction coefficient for J = 0.6

0.00-

. 9i ^ l u

\ Qi = 495 \ g i = 9 8 9

*- \ \ \ \ \

\ N \ \ N \ \ \ \ - ^ \ . v \ '

\ s 'Sv " ^

• ^ s

L i

Fig. 9(c) Traction coefficient for i = 1.8

£3,1 =£3,2= 1-83xlO"3, LJ?=1.63xlO-6, and g5 = 292. Subsequently, there are only four independent dimensionless groups which vary, namely , gu g2, £, and Lx. For most of the data to be presented, g2 is taken as 20.1, a constant, by using constant values of a and p0. A few data points are also obtained for g2 = 30.2.

The numerical results are shown in Figs. 4 through 9. The film thickness results are presented in the form of a thermal and non-Newtonian (Ree-Eyring) reduction factor, Cn which is defined as

Cr = H0/H0JN, (63)

where H0 is the dimensionless central film thickness calculated from the full thermal and non-Newtonian model, and H0jN is the dimensionless central film thickness calculated from an isothermal and Newtonian model. In this paper, HoiN is calculated from the central film thickness formula proposed by



Dowson and Toyoda (1978), which can be rewritten, in terms of the dimensionless groups in this paper, as

J f / 0 , / N=2.90^- 6V°- 0 6 . (64)

The reduction factor, Cn for gj = 240, 495 and 989, is shown in Figs. 4(a) and (£>). The results are obtained for £ = 0.1, 0.6, and 1.8, and for Z,, ranging from 0.081 to. 58.6, which covers most of the practical range of interest. As expected, the data show a decrease in Cr with an increase of the thermal loading parameter, Lu and an increase of the slide/roll ratio, £.

The data in Fig. 4 also show that Cr decreases with decreasing gi. One way of interpreting a decrease in g, is to increase the product of t/o and U0, with other dimensional quantities fixed. This product pan be changed while Lx is kept constant. Approximating 7 in equation (2) by %U0/h0, it can be seen that an increase of (TJ0 U0) will increase the shear stress and, therefore, the non-Newtonian effect. Thus, the decrease of Cr with decreasing gu results from a film thickness reduction due to increasing non-Newtonian effects.

It is interesting to notice that, especially for £ = 1.8, the effects of Ly on Cr decrease as gx decreases. This trend can be explained by observing that, for high values of (i}0 U0) and small values of Lu there exist appreciable film thickness reductions due to the non-Newtonian effects. As Lx increases, there are two processes taking place simultaneously. One is a film thickness reduction caused by the thermal effect, and the other is a gradual relief of the non-Newtonian reduction due to the decrease of shear stresses caused by the higher temperatures. As a result, increasing Lx does not give large reductions in film thickness, especially at high values of £ and small values ofgi.

For low slide/roll ratios, Fig. 4 shows an inverse " S " shape. This trend results from the fact that, for small Llt the fluid temperatures in the inlet region increase with increasing Lx. However, as Lt is further increased, this temperature remains almost constant. This behavior can be easily observed from temperature distribution plots which are not presented in this paper. On the contrary, for high slide/roll ratios, the inlet temperatures increase steadily, both for small and for large values of Lu and, therefore, no inflections are seen in the curves.

By performing a regression analysis on the data shown in Fig. 4 and a few data for a higher g2 value of 30.2, the following empirical formula is obtained for the thermal and non-Newtonian (Ree-Eyring) film thickness reduction factor, Cr.

CV= 1-0.2427, (65)

where Y is the thermal and non-Newtonian (Ree-Eyring) influence parameter, defined as

y = j L 0 . 2 6 z l . 7 g f 0 . 5 8 g 2 _ ( 6 6 )

In equation (66), Le is the effective thermal loading parameter, defined as £ e = l + 3 .08xl0- 6«fc„ (67)

and Z is the slip parameter, defined as

Z=max[t/1 > t / 2 ) / C / 0 = l + l £ l / 2 , (68)

where Ux and U2 are assumed to be positive. Equation (65) is valid over wide ranges of Lx and £. The

applicable range for g2, however, is limited to 20<g 2 <30 or nearby values, due to the small number of data points used in the regression analysis.

The influence of gt on the slope of the Cr-Lx curve has been included in the empirical formula by means of the dependence of Le on gu as given by equation (67). The form of both Le and Z have been so chosen that, as £ or Lx approaches zero, equation (65) gives a non-zero film thickness reduction, which is a condition required by the numerical data. For example, as Lx approaches zero, the non-Newtonian effect gives a film thickness reduction even though the thermal effects on film thickness are negligible.

With an increase in g2, equations (65) and (66) indicate a decrease in Cn i.e., a further reduction of film thickness. This is because a higher viscosity resulting from the higher Hertz pressure, p0, gives a higher shear stress, which enhances the non-Newtonian behavior.

The reduction factor, Cn given by equation (65), is plotted against the influence parameter, Y, in Fig. 5, with the numerical results shown as small circles. The maximum relative error in Cr between the line and the data points is 10 percent. The data points show a-one-sided deviation from the line as Cr approaches unity (isothermal, Newtonian case). This deviation results from the fact that Cr is forced to be unity as Y approaches zero in the regression process.

The results for the minimum-to-central film thickness ratio, Hm{JH0, are shown in Figs. 6(a), ib), and (c) for £ = 0.1, 0.6 and 1.8, respectively. A decrease of this ratio with an increasing Z,! results from an increasingly sharp drop of pressure at the outlet of the contact. For high £ and low gu the curves show decreasing slopes at large values of Lx. For these conditions, this trend can be explained by noting that, as shown by the temperature and pressure distributions which are not presented, the major increase of temperature with the increasing Lt occurs near the outlet, and, therefore, does not significantly affect the pressure distribution, which determines the film shape. For low £ and high gu however, as Z,, increases, the major temperature increase occurs in the central region of the contact, which changes the pressure distribution, giving a rapid decrease of Hmin/H0.

The ratio, Hmia/H0, decreases with an increase in gj except for low £ and Lx where the reverse is true. This ratio also decreases with an increase in £ except for low g, and Lx where the reverse is true. These two effects are caused by the presence of a pressure spike near the outlet for low £ and moderate to high g] values. For low glt the pressure distribution deviates appreciably from the Hertzian shape, and a high single pressure maximum appears near the center of the contact instead of a pressure spike near the outlet. The dependence of Hmin/H0

upon £ and g{ is significantly affected by the presence or absence of the pressure spike.

Although all the solutions have been obtained for a bulk temperature, T0, of 90°C, the results for the dimensionless film thickness, H0, can be applied to conditions with different T0 because H0 depends on dimensionless groups that do not directly involve T0. If distinct lubricants at different bulk temperatures have identical viscosity, J?0>

a n d other properties, the resulting H0 will be the same.

The midlayer temperature, Tm(x), is defined as the temperature at z = h/2. The maximum midlayer temperature, T'm.max- is plotted versus Lt in Fig. 7. Note that Tm max is not necessarily the maximum temperature for the whole flow field. The maximum surface temperature is plotted versus Lx in Fig. 8. As expected, for all the results, the maximum surface temperatures occur at the slower surface, which is surface 2 in this paper and is, therefore, denoted by T2mm. Although in this analysis a dimensionless temperature defined as T* = /3 (T— T0) was used, Figs. 7 and 8 show temperatures in their dimensional form (for r 0 = 90°C). Both Tm<max and T2tItax are observed to increase with an increase of L\, £, or g,.

To estimate the sensitivity of the presented results to changes in the Roelands temperature parameter, 8R, this parameter was

, varied by changing T0 for the operating condition specified by gi = 495, g2 = 20.1, £ = 0.6 and Z,, = 1.38 (a medium range case). The bulk temperature, T0, was changed from 90 °C to values of 0°C and 180°C. The viscosity, r/0, is not changed for this computation, implying that different lubricants are used to maintain constant values of gx and Lx when the Roelands temperature parameter, BR, is changed. Compared to the solution for T0 = 90 ° C, the changes in H0 and Hmin/H0 are less than one-percent. Furthermore, the change in the maximum midlayer temperature rise, Tmmax- T0, and the change in the

240/Vol . 113, APRIL 1991 Transactions of the ASME


in the traction coefficient are found to be less than six-percent. These results suggest that the influence of 8R on the film thickness, temperature rise, and traction coefficient is not significant.

In engineering practice, a change in the bulk temperature, T0, for a specific lubricant, gives changes not only in 6K, but also in g{ and Lx due to the strong dependence of viscosity, •ijo, on temperature. Numerical solutions were obtained about the same medium range case with the bulk temperatures changed from 90°C to 60°C and 120°C. Under these conditions, gi and Z,] will change.while g2 and £ will remain at 20.1 and 0.6, respectively. The film thicknesses obtained lie well within the maximum band of error for equation (65). The temperature rises, T„hmax - T0, for these two cases are consistent with the data presented in Fig. 7, corresponding to the values of gx and Lx associated with the values of T0.

The discussion above suggests that the temperature results shown in Figs. 7 and 8 can be used to predict maximum temperatures for conditions where T0 is other than 90 °C provided that the dimensionless temperature, V, is utilized.

For all the plots presented, the material-geometry parameter, LR, is held constant at 1.63 x 10~6. A sensitivity test on this parameter has been made for the medium range case mentioned above. When LR is decreased to 4.2 x 10~7, with all other dimensionless groups kept constant, no significant change is found in the numerical results. However, when LR is further decreased to 2.6 x 10"8, appreciable increases in the maximum midlayer temperature rise, T m m a x - T0, the maximum surface temperature rise, T2inax—T0, and the traction coefficient are found; the dimensionless central film thickness, H0, and the dimensionless film thickness ratio, Hmin/H0, are not significantly affected.

In order to understand the temperature behavior shown in Figs. 7 and 8, the energy equation, (32), will be examined. Since the lubricant film is very thin compared to the width of the contact, the convection term on the left-hand side of equation (32) is much smaller than the conduction term (the first term on the right-hand side). By neglecting the convection term, equation (32) can be rewritten as

(69)

Defining a non-Newtonian factor, Q„, as the ratio of the shear stress of a non-Newtonian fluid to that of a Newtonian fluid under the same rate of shear strain, y, i.e., Q„ = r/(riy), the dissipation function, $, can be rewritten as follows:

<!> = T-Y = QnLit;2(y/Vo)

(du/dz)2

M-Utf/h* k0/(W

2)- (70)

The behavior of the temperature near the center of the contact can be easily explained by the use of equation (70). For this region, where dp/dx is relatively small, the quantity in the square brackets in equation (70) is around unity. Noting this fact, and substituting equation (70) into equation (69), a simplified energy equation in a nondimensional form is obtained

d2T* + QnL&2(rl/i)0)(k0/kf)=0. (71)

It is obvious from equation (71) that, as Lx or £ increases, the heat generation term increases, and, hence, the temperature increases. The factor, (17/170), serves as a damping factor in the sense that it becomes smaller when the temperature is increased. The factor, (k0/kf), depends on pressure, as given by equation (41). The effects of g{ on temperature can be explained by considering the case with all dimensional quantities fixed except for r\0 and U0. Under these conditions, as gi increases, (rjoUo) decreases, and Q„ increases as a result of decreasing (1)7). This increase in Q„ indicates a decrease in non-Newtonian effects.

For a constant L^2, this gives an increase in the heat generation term, resulting in an increase in the temperature. Consequently, Tm.max increases with increasing gu as shown in Fig. 7. Moreover, from Fig. 8, T"2max also increases with the increase of gu

since the amount of heat transferred to the solids depends on the temperature gradients, whose magnitudes are larger for a higher midlayer fluid temperature.

The traction coefficients, f, versus Lx for g{ = 240, 495, and 989 are plotted in Figs. 9(a), (b), and (c) for £ = 0.1, 0.6, and 1.8, respectively". As can be seen from Fig. 9(a), for small £ and small Luf increases slightly with increasing Lx for both g! = 495 and 989. This increase in traction coefficient is caused by a decrease in the film thickness due to a slight temperature rise in the inlet region. For these conditions, the temperature rise in the contact zone does not significantly affect the shear stress. For higher Lx and £, however, the temperature rise significantly reduces the shear stress, giving a decrease in / with an increase of Lx.

For low £ and Lu the traction coefficient is observed to decrease with decreasing glt which implies an increase in (ri0U0) when other dimensional quantities are fixed. This is because an increase in (r/0t/0) makes the pressure distribution deviate significantly from the Hertzian shape and shifts the distribution toward the inlet. As a consequence, the viscosity is relatively low in the region near the outlet since the pressure is relatively small. This pressure shape gives a shear stress distribution over a small effective domain, and, therefore, gives a small traction coefficient. For moderate to high £, however, the traction coefficient increases with decreasing gu which implies an increase in (7/0£/o) when other dimensional quantities are fixed. This is because, for a high £, the shear stress is strongly affected by the factor (i?f/0£) rather than by the pressure distribution.

By comparing Fig. 9(a) with (b), and (b) with (c), it can be seen that the traction coefficient increases with increasing £ for small Lx (nearly isothermal conditions), and decreases with increasing £ for higher Lx which implies significant thermal effect. This is because an increase in £ gives a large increase in temperature whenZ,,£2 is high, and this temperature increase reduces the viscosity, and, hence, reduces the shear stress in the contact. These trends are consistent with the behavior of the experimental traction curves obtained by Johnson and Cameron (1967-1968). Their experimental curves show that, for a range of nearly constant U0 and ij0, the traction coefficient increases with increasing £ for low £ (small £j£2), and decreases with increasing £ for high £ (large Z-i£2).

In order to ascertain the effect of the pressure-dependent thermal conductivity, kj- given by equation (41), a case for g{ = 629, g2 = 20.1 and £ = 0.6 has been investigated. By using a constant kf= k0, the maximum midlayer temperature, Tm max, is found to be 320°C, and the maximum surface temperature, T̂ maxi is found to be 137°C. By using the pressure dependent kf, Tmmax is decreased by 27°C and T2max is increased only by 1°C. Therefore, the major effect of the variable kj is on the maximum temperature in the fluid.

It is natural to try to compare the results presented in this paper with those from similar analyses in the literature, among which are the results of Wilson and Sheu (1983), which give much smaller Cr values for high Lt than the present results since only conduction was considered in their analysis. In fact, convection is important in the inlet region, especially for high speeds, which are usually associated with high values of X,. To illustrate this fact, consider the following results based on g{ = 495, g2 = 20.1, £ = 0.6 and L, = 1.38. The heat transferred by conduction to both surfaces in the region from x= — 3a to the stagnation point is found to be only thirty-six percent of the total heat generated in this region, with the rest of the heat being transferred by convection, of which thirty-six percent is transferred downstream and twenty-eight percent is transferred out of the contact by back-flow.



Conclusions

A formula for the thermal and non-Newtonian (Ree-Eyring) film thickness reduction factor, C„ has been obtained by regression based on numerical results. This formula, equation (65), shows a decrease in Cr with an increase of Lu £, and g2, and with a decrease of gi. The change of the reduction factor, Cn with Ll is influenced by the value of glm The formula also shows that, even when the thermal loading parameter, Llt

approaches zero, there still exists a film thickness reduction due to non-Newtonian effects. The solutions presented are useful in the design of gears, rolling element bearings and cams where elastohydrodynamic line contacts exist.

The changes in the ratio, Hmin/H0, with g, and £ depend strongly on the changes in the shape of the pressure distribution. This ratio decreases with increasing g, except for low £ and Llt and decreases with increasing £ except for low gj and L\.

The maximum midlayer and surface temperatures are observed to increase with an increase in Lx, £, or g,. The maximum midlayer temperature in the lubricant is significantly affected by the dependence of lubricant thermal conductivity on pressure.

The traction coefficient is observed to increase slightly with an increase of Lx for low £ and Lu and to decrease with an increase of L, for all other conditions.

Acknowledgment The authors wish to express their appreciation to the Na

tional Center for Supercomputing Applications and the National Science Foundation for allocating computing time for this research on the CRAY X-MP/48 supercomputer at the University of Illinois at Urbana-Champaign.

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D I S C U S S I O N

Scott Bair1 and W. O. Winer1

The inclusion of the energy equation in any non-Newtonian analysis of elastohydrodynamic lubrication is an important step. The rheological and thermal properties of the liquid are dependent upon local temperature. The discussers are, however, puzzled by the rheological model selected by the authors. The logarithmic shear stress nature of the Eyring model with

'George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405.

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Grubin, A. N., and Vinogradova, I. E., 1949, "Investigation of the Contact of Machine Components," Central Scientific Research Institute for Technol. and Mech. Engr., Book No. 30, Moscow (D.S.I.R. Translation, No. 337).

Hahn, S. J., Eyring, H., Higuchi, I., and Ree, T., 1958, "Flow Properties of Lubricating Oils Under Pressure," NLGI Spokesman, Vol. 22, June, pp. 121-128.

Johnson, K. L., 1970, "Regimes of Elastohydrodynamic Lubrication," J. Mech. Engr. Science, Vol. 12, No. 1, pp. 9-16.

Johnson, K. L., and Cameron, R., 1967-1968, "Shear Behaviour of EHD Films at High Rolling Contact Pressure," Proc. I. Mech. E., Vol. 182, Part 1, pp. 307-319.

Johnson, K. L., and Tevaarwerk, J. L., 1977, "Shear Behaviour of EHD Oil Films," Proc. Royal Soc, London, Series A, Vol. 356, pp. 215-236.

Martin, H. M., 1916, "Lubrication of Gear Teeth," Engineering, London, Vol. 102, p. 119.

Ree, T., and Eyring, H., 1955, "Theory of Non-Newtonian Flow. I. Solid Plastic System," and "Theory of Non-Newtonian Flow. II. Solution System of High Polymers," J. Appl. Physics, Vol. 26, No. 7, pp. 793-800, (Part 1) and pp. 800-809 (Part II).

Richmond, J., Nilsson, O., and Sandberg, O., 1984, "Thermal Properties of Some Lubricants Under High Pressure," / . Appl. Physics, American Institute of Physics, Vol. 56, No. 7, pp. 2065-2067.

Roelands, C. J. A., Vlugter, J. C , and Waterman, H. I., 1963, "The Viscosity-Temperature-Pressure Relationship of Lubricating Oils and Its Correlation with Chemical Constitution," ASME Journal of Basic Engineering, Vol. 11, pp. 601-610.

Sibley, L. B., and Orcutt, F. K., 1961, "Elasto-Hydrodynamic Lubrication of Rolling Contact Surfaces," ASLE Transactions, Vol. 4, No. 2, pp. 234-249.

Spalding, D. B., 1972, "A Novel Finite Difference Formulation for Differential Expressions Involving Both First and Second Derivatives," International Journal for Numerical Methods in Engineering, Vol. 4, pp. 551-559.

Wang, S. H., and Zhang, H. H., 1987, "Combined Effects of Thermal and Non-Newtonian Character of Lubricant on Pressure, Film Profile, Temperature Rise, and Shear Stress in E.H.L. ," ASME JOURNAL OF TRIBOLOGY, Transactions ASME, Vol. 109, pp. 666-670.

Wilson, W. R. D., and Sheu, S., 1983, "Effect of Inlet Shear Heating Due to Sliding on Elastohydrodynamic Film Thickness," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 105, No. 2, pp. 187-188.

properties averaged over the entire contact corresponds well with the average shear stress observed in EHD traction. However, when applied locally as a rheological equation of state, it fails to represent the most pronounced non-Newtonian effect observed in primary measurements—that of a rate independent limiting shear stress (Bair and Winer, 1979, 1982, 1990 and Ramesh and Clifton, 1987). It is interesting to note that the authors reference our 1979 paper showing limiting shear stress behavior for EHD lubricants.

This limiting stress has been observed in concentric cylinder rheometers (Bair and Winer, 1979, 1982, 1990) with a pres-surization time on order of minutes, and impact pressure shear plate experiments (Ramesh and Clifton, 1987) for which the pressurization time was a few hundred nanoseconds. Indeed, the model oil in this paper, LVI260, has been thoroughly in-



vestigated in this laboratory (Bair and Winer, 1982 and 1990). Newtonian behavior was observed to shear stresses of about 20 MPa (nearly ten times the required Eyring stress for traction curve fitting) followed by rate independent yielding at higher stresses. In the 1990 paper, lower pressures and higher shear rates were utilized in an attempt to find Eyring behavior. But, the transition from Newtonian to limiting shear was even more abrupt than at higher pressures. No primary evidence exists which supports Eyring's Sinh law behavior in lubricants at high pressure. Why then has so much time been spent incorporating it into an EHD analysis?

Additional References

Bair, S., and Winer, W. O., 1982, "Some Observations in the High Pressure Rheology of Lubricants," ASME JOURNAL OF LUBRICATION TECHNOLOGY, Vol. 104, No. 3, pp. 357-364.

Bair, S,. and Winer, W. O., 1990, "The High Shear Stress Rheology of Liquid Lubricants at Pressures of 2 to 200 MPa," ASME JOURNAL OF TRIBOLOGY, Vol. 112, No. 2.

Bair, Scott, 1990, "High Shear Stress Rheology of Liquid Lubricants," PhD thesis, Georgia Institute of Technology.

Ramesh, K. T., and Clifton, R. J., 1987, "A Pressure Shear Plate Experiment for Studying the Rheology of Lubricants at High Pressures and High Shearing Rates," ASME JOURNAL OF TRIBOLOGY, Vol. 109, No. 2.

Authors' Closure The authors would like to thank Dr. Bair and Professor

Winer for their discussion. The use of the Ree-Eyring fluid model in elastohydrodynamic (EHD) lubrication is firmly based on experimental evidence obtained on disc machines over a wide range of pressures and temperatures. The authors are confident that the results presented in this paper represent the best available estimate of the effect of heat generation on film thickness, temperature, and traction in an EHD contact. The Ree-Eyring model has a sound basis in the thermal activation theory of viscous flow. The parameters in this model, r0 and •q, can be expressed in terms of fundamental thermodynamic properties (Evans and Johnson, 1986, and Eyring, 1936), as follows:

T0 = kBQ/vT (A-l)

and

1 fE+VrP\

^ ' e ) = ^ e x p U ^ > (A-2) where kB is the Boltzmann constant, 9 is the absolute temperature, C is a constant determined by the microscopic structure of the fluid, E is the thermal activation energy for the flow, and vp and vT are the activation volumes for pressure and for shear stress, respectively. The Ree-Eyring model is valid when the work done by the shear stress in promoting flow, VTT, is small compared to the total activation energy, E+pvp. When the former quantity is comparable with the latter, the fluid behavior must be described in some other way. Experiments conducted by Imai and Brown (1976) on amorphous solid polymers suggest that the mechanism of flow changes from thermally activated motion of independent molecular segments to the formation of shear bands through the collaborative motion of adjacent segments when the shear stress reaches the order of 1/30 of the elastic shear modulus. The material can deform plastically under a constant shear stress— a limiting shear stress—when shear bands are formed.

The experiments by Bair and Winer (1979, 1982, and 1990) indicated that lubricating fluids also reach a limiting shear stress. Experimental data obtained by Evans and Johnson (1986) showed that values of the limiting shear stress were on the order of 1 /45 of the elastic shear modulus of the lubricants. This suggests that the limiting shear behavior might be caused by the formation of shear bands or a similar mechanism, other

than thermal activation, and that the limiting shear stress represents the ultimate strength of the material.

In Bair and Winer (1990), the limiting shear stress data for LVI-260 mineral oil indicated values of rL/p ranging from 0.031 to 0.040, while the value of 0.047, given in their Table 2, was used for comparison with the data of Johnson and Tevaarwerk (1977). A close examination of the models used to interpret their data from the high-pressure couette viscometer reveals a sensitivity of the experimental results to thermal effects intrinsic to the apparatus, which cannot be directly measured. This is especially true for the data obtained with very small gaps between the concentric cylinders. Moderate temperature rises at the stationary inner surface could serve to decrease the gap (thus increasing the real average shear strain rate) and reduce the viscosity in the vicinity of the inner wall (further increasing the local shear strain rate near the inner wall), while the perceived average shear strain rate would remain unchanged. This is a possible explanation for the apparent Newtonian behavior observed by the discussors for some fluids.

Bair and Winer (1979) proposed the following limiting shear stress model:

7 = - — l n ( l - T/TL) f o r r > 0 . (A-3) V

As the shear strain rate, 7 , tends to infinity, equation (A-3) shows that the shear stress, r, approaches the limiting shear stress, TL. This model gives a good description of the shear behavior near the limiting shear stress for some lubricants. However, in this model, the initial non-Newtonian behavior at stress levels much lower than the limiting shear stress is determined by the value of the limiting shear stress. Since it is likely that the initial non-Newtonian behavior of a lubricant originates from a physical mechanism that is different from the limiting shear stress, the following rheological model would be more appropriate.

f77 sinh"1 ( IJ7/T0) for 1177 I <r0 sinh (TL/T0), T= (A-4)

(jL for I ij71 > T, smh {TL/T0).

Equation (A-4) is a generalization of the model proposed by Jacobson and Hamrock (1984). Viscoelastic effects are not included under the assumption that the Deborah number is usually below one for the operating conditions usually found in machines. Equation (A-4) should only be applied when coupled with a thermal analysis and would require data on the variation of TL with both pressure and temperature. Unfortunately, these data are only available for a few lubricants and for a very limited temperature range.

Since the Ree-Eyring model is valid only when the shear stress is less than the limiting shear stress, it is desirable to compare the calculated shear stresses to limiting shear stress values. The experiments to obtain the limiting shear stress values of LVI-260 oil (Bair and Winer, 1982 and 1990) were conducted in a temperature range from 20 to 35°C. In order to compare the numerical solutions obtained in this paper to these experimental values of limiting shear stress, the dimen-sionless results presented in this paper are applied to the LVI-260 oil at different temperatures. For each case studied, with some dimensional parameters specified, an »j0 value corresponding to the given dimensionless parameters, gi and L1(

can be found. In order to obtain this r\0 value, the oil is considered to be placed at a temperature, T„, given by a viscosity-temperature relation at ambient pressure. In this way, each case studied would correspond to an operating condition with the LVI-260 oil at a bulk temperature, To, and at an average contact temperature, T„, approximately given by T„= To + &Tmymax/2, where an estimate of the maximum midlayer temperature rise, A7^m a x , can be obtained from Fig. 7 by using



the equation, A7^max= 7^max-90°C. An examination all the data for g2 = 20.1 showed that five cases had T„ values that are approximately within the temperature range tested by Bair and Winer. For these cases, the limiting shear stress is estimated as a function of the contact position, x, by using the formula,

TL = TLo + mp, (A-5)

where m = 0.047 (Bair and Winer, 1990), and TLo = 2.3MPa (Jacobson and Hamrock, 1984). The maximum value of T/TL for the whole contact is obtained for each case. This maximum ratio is found to be in the range from 0.61 to 0.81 for the five cases considered, indicating that the limiting shear stress values would not be reached. Thus, the initial assumption—that the governing constitutive equation (2) would be used, ignoring the effects of limiting shear stress—appears to be correct.

Bair and Winer (1990) argue that the apparent Ree-Eyring behavior in disc machine experiments is a consequence of the distribution of the boundary of a "plastic" region in the lubricant near the EHD contact center. This conjecture was based on an isothermal analysis with a model that assumed Newtonian behavior when the shear stress was below TL (see equation (17) of the above reference). Yet Bair and Winer (1979) postulated a constitutive equation (A-3) based on their experimental observations—that non-Newtonian effects are appar

ent at shear stress levels below the experimentally observed limiting shear stress levels. If this equation were used, the "plastic" central region would occur at much larger shear strain rates, or it might not occur due to the initial non-Newtonian and thermal effects (for T < TL) as are noted in this paper. When shear strain rates are large, thermal effects must be considered.

The benefit of this discussion is to point out the need for corroboration of the limiting shear stress data of the discussors and to underscore the need for robust experiments to help resolve the uncertainty about the actual physical (and/or chemical) behavior of engineering lubricants at high pressures, temperatures, and shear strain rates.

Additional References Evans, C. R., and Johnson, K. L., 1986, "The Rheological Properties of

Elastohydrodynamic Lubricants," Proc. I. Mech. E., Vol. 200, No. C5, pp. 303-312.

Eyring, H., 1936, "Viscosity, Plasticity and Diffusion as Examples of Reaction Rates," /. Chem. Phys., Vol. 4, pp. 283-291.

Imai, Y., and Brown, N., 1976, "Environmental Crazing and Intrinsic Tensile Deformation in PMMA," J. Materials Science, Vol. 11, pp. 417-424.

Jacobson, B. O., and Hamrock, B. J., 1984, "Non-Newtonian Fluid Model Incorporated into Elastohydrodynamic Lubrication of Rectangular Contacts," ASME JOURNAL OF TRIBOIOGY, Vol. 106, No. 2, pp. 275-284.



thermal analysis of · for the ranges of parameters usually found in practice. by using a...

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