Wear, 60 (1980) 253 - 268 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

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GENERALIZED REYNOLDS EQUATION WITH SLIP AT BEARING SURFACES: MULTIPLE-LAYER LUBRICATION THEORY

J. B. SHUKLA, S. KUMAR and P. CHANDRA

Department of Mathematics, Indian Institute of Technology, Kanpur (India)

(Received August 28, 1978; in final form April 17, 1979)

Summary

A generalized form of Reynolds equation for fluid lubrication has been derived considering the effects of viscosity variation in the film and slip at the bearing surfaces. Various specific cases have been deduced and the con- cept of multiple-layer lubrication introduced. The higher lubricant viscosity near the bearing surface is beneficial in reducing the coefficient of friction but the effect of slip is unfavourable.

1. Introduction

Generally most lubricated systems consist of moving (stationary) sur- faces (plane/curve, loaded/unloaded) with a thin film of an external material (lubricant) between them. The thin film between the surfaces supports the load and minimizes friction. Characteristics such as film pressure, surface frictional force and lubricant flow depend upon the nature of the surfaces, the nature of the lubricant film and the boundary conditions at the surfaces.

The equation governing the pressure generated in the lubricant film can be obtained by coupling the equations of motion with the equation of con- tinuity and was first derived by Reynolds [l] . In deriving this equation the thermal, compressibility, viscosity variation, slip at the surface, inertia and surface roughness effects were ignored. Cope [2] modified the Reynolds equation by including viscosity and density variation along the fluid film. The viscosity variation across the film thickness has been considered by Zienkiewicz [ 3,4] . Cameron [ 51 pointed out that the temperature gradient and viscosity variation across the film should not be ignored. Dowson [6] generalized the Reynolds equation by considering the variation of fluid prop- erties both across and along the fluid film thickness. Many workers have studied the effects of viscosity variation in lubricated systems by considering the Reynolds equation and an energy equation. A different approach which avoids the use of an energy equation in the study of viscosity variation was

254

proposed by Tipei and coworkers [ 7 - 91 and Quale and Wiltshire [lo] who assumed that there was a relation between viscosity and film thickness.

Very little attention has been paid to the study of the effects of slip at the surface, although it may be of importance in the flow behaviour of gases and liquids particularly when the film is thin [ 11 - 131, the surface is smooth [ 141 and at the porous boundary [ 15 - 201.

In gas bearing applications such as gyroscopes [ 211 the bearings operate in the slip flow regime. In this case the bearing film is either at a low pressure or extremely thin and the length of the molecular mean free path h becomes comparable with the film thickness h. The gas appears to detach from the sur- face at a finite velocity [ 11, 12, 221 . Increasing the temperature of the gas film further enhances this effect owing to the increase in the molecular mean free path [ll, 231.

To study the effect of slip Burgdorfer [24] modified the Reynolds equation for gas-lubricated hydrodynamic bearings operating in the slip flow regime under isothermal conditions and pointed out that if 0 < h/h < 1 the gas flow can be assumed to be continuous and the analysis can be carried out with a modified slip boundary condition. Hsing and Malanoski [ 251 studied the effect of the molecular mean free path in a spiral grooved thrust bearing and Tseng [23] used the Reynolds equation with slip to study the rarefaction effects of gas-lubricated bearings in a magnetic recording disc file.

The boundary conditions for slip flow at the surface of a gas bearing can be written as [ 11, 241.

(1)

where f is the reflection coefficient, h is the mean free path and u is a numer- ical constant. Because u and f are close to unity it can be assumed that ~(2 - f)/f is unity [24]. As the molecular mean free path X depends upon fluid viscosity, pressure and temperature it can be approximated by the relation [ 231

16 h z-_ ‘7 (RT)l’2

5(2#” p

where R is the gas constant, T the temperature of the gas, n the viscosity of the gas and p its pressure.

The effect of slip is also important on the flow behaviour of liquids especially when the bearing surface is very smooth and is operating at higher temperatures where the viscosity of the base oil decreases near the surface [26, 271. This effect has been studied for liquids [14, 26 - 281. The slip velocity at the wall can be written as

1 u,fip =- (3)

P

255

where 0 is the coefficient of sliding friction at the wall and q is the liquid viscosity.

The slip phenomenon also plays an important role in bearings with porous facings. Beavers and coworkers [ 15 - 171 discussed this effect for an incompressible fluid and demonstrated the existence of slip velocity at the porous surface. This has been further supported by Saffman [ 181, Taylor [ 191 and Richardson [ 201. The slip velocity at the porous surface can be written as

4 u2 au us&) = - -

t; ( 1 az wall

(41

where < is the slip coefficient at the wall and $ is the permeability of the porous facing.

In this paper a generalized form of Reynolds equation is derived for fluid film lubrication with slip velocities at the surfaces by considering the variation of fluid properties both across and along the film thickness.

2. Generalized Reynolds equation

The physical configuration of fluid flow between two curved surfaces is shown in Fig. 1. The basic equations of motion and the equation of con- tinuity for a Newtonian fluid considering the variation of fluid properties both across and along the film thickness can be written as

I(

au av _+- - 77 _._--

)I

+

ax ap

Dv Pg=PY-

a (6)

-+ at

; (PU) + ; bv) + .jf (PW) = 0

256

Fig. 1. Coordinate system.

With the usual assumptions of lubrication theory eqns. (5) can be simplified to [6]

ap a au -=- 77-

i 1 ax a.2 a2

ap a au -=- 77---

i 1 ay a2 a2

where p = p(x, y) is the film pressure. The boundary conditions considering slip at the surfaces are

u= (U)l =(A), 2 + u, i 1 1

au v = (V)l = (6 )l z i 1 + Vl

1

au u = (u)‘j =-(A), s ( 1 + u,

2

av v=(v), =-(6)2 g + v,

( 1 2

at2 =H,

(8)

atz=H,

where ( )1 and ( )z denote the value at z = HI and z = Ha respectively. Here the Xs and 6 s are the molecular mean free paths for gas lubrication and depend upon lubricant temperature, pressure and viscosity. With liquid lubrication X and S depend on viscosity and the coefficient of sliding friction. However,

251

with porous bearings h and 6 are functions of the slip coefficient at the wall and the permeability parameter of the porous facing (see eqns. (1) - (4)).

By integrating eqn. (7) and using boundary conditions (8) expressions for the fluid film velocities are obtained:

u = u1 + (a,H, + y) 2 t HI

+ u2 - Ul ( Fo --)(a1 + j ;,

Hl

where

H2

.z

Fb =PI +P2 + J d-a/r, H1

H2

F1 = a1H1 +a2H2 + J 2 dz/r, HI

(9)

(10)

H2

F; =&HI +P2H2 + J z w77

HI

@I,

a1 =Gx (6 )l

I31 =Gl

@I2

a2 =GL

2 s2 =s

2

Integrating the equation of continuity (eqn. (6)) with respect to z and taking limits from z = HI to z = H, gives

J H2 a H2$~+ J

H2 a

HI HI

a3c(w)d~+ J Hl

ay (PU)~ + WI:: = 0 (11)

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Simplifying eqn. (11) gives

The integrals of (pu) and (pv) are evaluated by partial integration. In- troducing the expressions for (pu) and (pu) and their derivatives in eqn. (12) gives

(13)

where

F2 = s”’ !+(+)& Hl 0

FL = j2 ;(z-so)dz

H1

F, = H2 pz

s -dZ

HI 77

259

Equation (13) represents a generalized form of Reynolds equation for com- pressible fluid film lubrication considering slip velocities at the bearing sur- faces. The two sets of functions F and G depend upon the variation of fluid properties both along and across the film and on the slip conditions at the surfaces.

In the case of no slip at the boundaries, i.e.

(X)1 = (A)2 = (h), = @)2 = 0

a1 = a2 =p1 =fl2 = 0

the generalized eqns. (13) and (14) reduce to

=H2 a+ (PU)Z + a+ (PU)Z 1 ---HI 1

a (u2 - Ul)(J’, + %I

-ax

- V1W3 + G2) +

FQ FO

where

F. = s”’ ddr,

HI

F2 = r” !-?(~---~)& Hl 77

(15)

F, = 7 PZ ddr,

HI

260

A generalized form of Reynolds equation derived by Dowson [6] can be deduced from eqn. (15) by taking HI = 0 and Hz = h. The various other forms of Reynolds equations can also be derived from the generalized eqn. (13).

3. Particular forms

3.1. Viscosity variation across the film The viscosity of the lubricant can vary across the film and may be dif-

ferent near the bearing surfaces owing to the reaction of additives and sur- factants with the surfaces [lo, 14,29 - 331. The most general form of Reynolds equation to study such a situation is given by eqn. (13).

Considering a reasonable case where the density and viscosity of the lubricant near the bearing surfaces may be different from that of the central region gives

P =Pl(x, Y) 7) =Bl@* Y) HI <z<H, +h,

P = P2(% Y) 17 =7)2(x, Y) HI + h < z < HI + h, + h2 (17)

P =Ps(x, Y) 7) =7?3@, Y) HI +hl +h2<z<H1 +hl+h2+h;

This introduces the concept of multiple-layer lubrication. By taking

u, = u us = VI = vz = 0

al=01 a2 =P2 (18)

aPi -= 0 az

i = 1, 2, 3

the generalized equation with slip reduces to

=H2 ;(Pu), p, 1 -Hl\ ;(PNl a; 1 + +-CPU), + -((Pv)1

261

(19)

where

F, =(yl +(yp +s+s+s 171 1)2 773

F1 = alHI + a2H2 + hl(ml +h) -+ hz(Wl +W +h2) +

2771 2712

+ h3(2H1 + 2hl + 2h2 + h3)

-2%

F2 = 2; {(HI + h1)3 --H!) + E {(H, + hl + h2)3 --- (H, + h1>3) +

FlF3 +3${H; -(HI +hl +h2)3) -F

3 0

F3 = $QH, +k) + +H, + 2hl + h,) + 1 2

+ 2h, + 2h, + h3) (20)

(W)l 'Pl&l

(Pv)2 =--p,a,(H2 -2j$

aH2 bwl$ = (PU)2 -fyi- +@u)2 a$ -

-(PU)l z aH1

-W)1 ay - vs

Here V, is the resultant velocity towards the film. To see the effect of slip consider three symmetrical incompressible

layers between two solid boundaries such that (see Fig. 2)

7)1=172 PI =P2=P3

+ ~~~~~ L ~ ~~ -l Fig. 2. Slider bearing with three layers of lubricant.

H, = 0 H, =h+H=h, h, =h, =H/2 h, =h (21)

a1 =a2 = PI = 02 = l/P

The appropriate Reynolds equation can be written from eqn. (19) as

(22)

where

F 4

_(h,-W3 +h,3-(k-W3 h2 +”

120, 1277, P

(23)

p is the coefficient of sliding friction (see eqn. (3)). By using eqn. (22) for a one-dimensional slider bearing the equation

governing the film pressure in the dimensionless form can be obtained as

where

jY,=__= TJ2F4 (6, --H)3 +i;z -(ix -ii)3 +EL

hz2 12 12; -fi

p _ ph:,

772UL

$!L @x2 - x p=_.- x=-

7)2 772 L

(24)

(25)

Integrating eqn. (24) with the boundary condition

263

i O 6Loo

1

01 02 03 -

1/p --

Fig. 3, Variation of X0 with slip coefficient.

6 h

z= 0 &&X0, il, =h,o =x0 h x2

gives

di 1 _ s = 2~, (hx -ix,) (26)

which after integration and using the conditions 6 = 0 at 2 = 0 and 1 gives

p=l go 1 -

J

Gl-

20

FT(hx --I;,,)+; 1 Fg(ko -h,)d? X0

where F. is given by

To 1 _ s 0

FY(h, --tE,,)dz-- j &fix0 -&.)d;=O CO

(27)

(28)

Equation (28) is numerically solved by considering the film thickness function

h=h,. --ax

where cr is the angle of inclination of the slider. The solution is plotted in Fig. 3 for various values of the slip coefficient l/p and viscosity ratio 7, where it can be seen that To decreases with increasing slip coefficient but increases with 11 for lower values of l/j?. Since x0 gives the location of maximum pres- sure in the film it shifts towards the diverging side of the slider bearing owing to slip effects.

The dimensionless load capacity, friction force and coefficient of fric- tion are obtained by following the usual procedure and using eqns. (26) and (27):

264

02

HZ01

0 15

T I

k,22 hx2

E L /.4='s cci,l 2 0, hx2

005 'I='0

i=O5

O ooLr% 0’2 0’3 0’4 1/p +

Fig. 4. Load capacity.

Fig. 5. Friction force.

w = Wh,2, ;o T(h, _=- r 772UL 0 a

--LOldz+ r l 3C(h,() -i;,, dn:

2F, (29) CO

WLO A)

4F, t d3t

c,=im

(30)

(31)

where

712~0 F, = - Fz 2

h x2

+H+~+~

i;, =tEJc1 -cux: Cl = k,lhxz G = &L/h,,

The expressions for w, F and c, are integrated numerically and plotted in Figs. 4, 5 and 6 respectively. The load capacity and frictional force decrease from their values with no slip (1 /p + 0) as the slip coefficient l/p increases but increase with increasing 6. Figure 6 shows that the coefficient of friction decreases as 7, increases and hence the increase in load due to 7, is more pronounced than that of the friction force which leads to a lower coefficient of friction. Thus the higher viscosity of the lubricant near the surface is beneficial in the hydrodynamic lubrication process with or without a slip condition and this condition may exist when additives or surfactants are present in the lubricant [ 141. Figure 6 shows that the coefficient of friction increases as l/p increases and therefore the decrease in load is more signifi- cant than that of friction.

265

Fig. 6. Coefficient of friction.

3.2. Viscosity variation along the film The Reynolds equation for a two-dimensional slider bearing with a

single layer of compressible lubricant which has varying properties along the film, i.e.

P = P(X, Y) 77 = 7)(% Y)

Ql =P1 a2 =P2

u, = u I!72 = v, = v2 = 0

can be deduced from eqn. (13):

where

F0 = a1 + a2 + H2 -- HI

rl

F, =cu,H1 +cv2H2 + H$ -H;

277

FE =P H; -Ht

377 - + qH,2 + cx2H;

Taking

HI = 0 H2 =h

011 = a2 = 01 = P2 = h/77

(32)

PV (33)

(34)

(35)

266

Fig. 7. Hydrostatic bearing with three layers of lubrication.

Fig. 8. Pressure distribution in the hydrostatic bearing: relri = 20.0.

gives

which is the same equation obtained by Burgdorfer [ 241 when the squeeze velocity V is zero.

Consider the flow of a gas with constant viscosity for an externally pressurized hydrostatic bearing as shown in Fig. 7. Reynolds equation can be written from eqn. (36) as:

(37)

where pa and A, are taken as the reference values in the bearing recess OS r< ri.

Following the usual method of integrating eqn. (37) and using the boundary conditions

P=Pa at r = ri (33)

p=o at r = re

the film pressure can be obtained as

(39)

To see the effect of A, on p/p, we examine

267

(40)

Since

ln(r,/r) < 1 -- ln(r, hi 1

the expression inside the square brackets in eqn. (40) is positive. Hence (d/d&&p/p,) is negative. Thus it can be concluded that the pressure decreases as the molecular mean free path X, increases. This effect can also be seen from the plot of eqn. (39) given in Fig. 8 for various values of ha/h. Since the load capacity is the integral of the pressure it also decreases as h, increases.

4. Conclusion

A generalized form of Reynolds equation applicable to fluid film lubrica- tion was derived by considering the variation of fluid properties both across and along the film thickness with slip velocities at the bearing surfaces. A one-dimensional slider bearing with three symmetrical incompressible lubrica- tion layers was studied and the beneficial result for hydrodynamic lubrica- tion due to the presence of increased viscosity near the bearing surface was indicated. However, although the effect of slip at the bearing is to decrease both the friction force and the load capacity, the coefficient of friction in- creases which leads to unfavourable results.

For a gas-lubricated hydrostatic bearing the gas film pressure and load decrease with increasing molecular mean free path.

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