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Lubrication of rough surfaces—A microcontinuum analysis
Int. Z Mecg $cL Vol. 24, No. 10, pp. 619-633, 1982 Printed in Great Britain.
0020-7403/82/100619-15503.0010 © 1982 Pergamon Press Ltd.
LUBRICATION OF ROUGH S U R F A C E S - A MICROCONTINUUM ANALYSIS PRAWAL SINHA a n d CHANDAN SINGH Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur-208016, India SummarykNumerous papers have been devoted to the study of the influence of surface roughness on bearing systems. These studies, however, do not account for the non-Newtonian situations which arise so frequently in present day lubrication practice. One way of studying these non-Newtonian effects is through microcontinuum analysis, wherein the micromotions and deformation of fluid microcontinua are considered. These microscopic effects may be of great significance when the surfaces are rough. The micropolar fluid theory, in which the deformation of the microelement is ignored, is therefore applied to study the effects of surface roughness in lubrication. The problem considered is that of a rough journal bearing, where, for convenience, only the stationary part of the bearing is assumed to be rough. In line with Christensen's stochastic theory, the cases of longitudinal and transverse roughness are studied. The results obtained are in qualitative agreement with Christensen's results. However, it is shown that the effects of surface roughness are more pronounced, when analysed from a microcontinuum view point. NOTATION A surface area b half total range of random film thickness variable B b/c C bearing clearance CR coefficient of friction parameter E() expected (mean) value of () f frictional drag fO probability function f(N, a, h) defined by equation (4) F dimensionless frictional drag FR frictional drag parameter S(N, a, h) defined by equation (25) G(N, L, H, B) defined by equation (58) h h = h. + h,, film thickness for rough bearing h. smooth part of film thickness h, random part of film thickness h.o, h.~ defined by equations (35) and (48), respectively H, Ho, Ho. H, defined in equation (49) a characteristic material length, (-//4~) 112 L length ratio, c/a N coupling number (~,1(2/~+ 7,))"2 P hydrodynamic pressure, in general, a random variable P dimensionless pressure q . q~ fluxes in x and y directions, respectively R journal radius &(N, L, H, B) defined by equation (57) S~(N, L, H, B) defined by equation (64), i = 2, 3, 4 t time U tangential journal velocity Ww/2,W0 load capacities perpendicular and along the line of centers, respectively W resultant load capacity W,~2, W0, W dimensionless forms of w.12, w0 and w, respectively NR load ratio parameter X, y cartesian coordinates viscosity coefficients for micropolar fluids E eccentricity ratio 0 angular coordinate, x = RO dynamic viscosity coefficient of friction random variable I" shear stress A bar written above a variable denotes the mean or expected value of the corresponding variable. 619
620
P. SINHA and C. SINGH INTRODUCTION
The analysis of surface roughness effects on lubrication has attracted considerable attention and has become a subject of intensive study during the last two decades. This is due to their importance in the practical field, their physical significance and also their great theoretical interest. Most theoretical studies on bearing lubrication appearing in the literature make use of the assumption that the bearing surfaces are perfectly smooth. It has long been recognized that owing to machining limitations, this assumption is rather unrealistic, particularly in bearings, where the film thickness is small. In recent years a major way of analyzing the circumstances has been through the adoption of statistical methods. Two basic approaches have been developed, namely the deterministic and the stochastic approach. The starting point in each case has been the Reynolds equation. In the deterministic approach the surface roughness was accounted for by postulating a sine (cosine) wave or a series of sine (cosine) waves for the film thickness[I]. More recently Tzeng and Saibel[2, 3] have used stochastic concepts to study the surface roughness effects in slider bearings[2] and in short journal bearings[3]. The analysis was concerned with the problem of a two-dimensional slider bearing, with one-dimensional roughness in the direction transverse to the sliding direction. There is, however, experimental evidence [4] which often exhibits a one-dimensional type of roughness running in the direction of sliding, generally referred to as longitudinal roughness, Fig. l(a). Uni-directional roughness has also been observed and studied by Moore [5]. In line with this evidence, Christensen and Tonder[6, 7] and Christensen[8] presented a stochastic model for the hydrodynamic lubrication of rough surfaces. Christensen[8] considered two types of one-dimensional roughness, the longitudinal roughness, Fig. l(a), with its striations parallel to the sliding direction, and the transverse roughness, Fig. l(b), with its striations perpendicular to the sliding direction, and obtained a modified Reynolds equation by making two heuristic assumptions about the pressure gradient and flow rate. It was concluded that the longitudinal roughness results in a slight decrease in load-carrying capacity and an increase in frictional force thereby causing a significant increase in coefficient of friction. The effect of transverse roughness is, however, to improve the bearing characteristics. Later Elrod[9] improved Christensen's modified form of Reynolds equation. The theoretical results obtained by using Christensen's theory have recently been verified experimentally by Jackson and Cameron[10]. Scores of papers have appeared in the wake of these earlier studies on various aspects of the lubrication of rough surfaces. All of the works which have appeared to date show that the surface roughness may have significant effect when the minimum film thickness is comparable to the roughness mean height. The effect is much pronounced for lower values of film thickness. Inherent in these studies was a fundamental assumption that all the lubricants behaved as if they were Newtonian. y
Y j
.
j
......
x
~z
Fig. l(a). A schematic diagram of longitudinal roughness.
~/1111111/11111111
El- u-- F"
ih:x
I
I
/I/I//1111IIIIIIII/
l"
~Z
Fig. l(b). A schematic diagram of transverse roughness.
Lubrication of rough surfaces--a microcontinuum analysis
621
Considerable experimental evidence exists, Henniker [ l l], which however indicates that, fluids subjected to small clearances, such as those found in lubrication, may behave in a non-Newtonian way. Needs[12] also obtained results, which indicated a non-Newtonian behaviour in circular squeeze films. In a later review, Hayward and Isdale[13] concluded that rheological abnormalities in pure liquids do not extend to more than a few molecular diameters from a solid boundary. It was also indicated that the inherent surface roughness may be a cause for rigidity observed by the earlier workers. A remarkable result is that of Askwith et al. [14] and Cameron and Gohar[15] who report that there is a plastic layer of a rigidity sufficient to arrest completely the descent of a flat metal plate, at a depth of c a 1.9/~m (-~x 10-4) inch, even in pure cetane. Kannel et al.[16] observe nearly a 100% elasto-hydrodynamic film thickness enhancement when anti-wear additives are present as compared with the base fluid. The film enhancement has been reproducibly observed with synthetic paraffin base fluids as weU[17]. Similar results have been observed by Draugiis et al.[18] and
Bollani[19]. The overall conclusion was an increased effective viscosity in thin films. Such thin films in which the properties are no longer those of the bulk fluids, are of great technological significance, since most practical machine elements owe their performance to them, especially in the boundary or the mixed lubrication region. Non-Newtonian situations may also arise in bearings where additives are used. The use of additives these days is a common practice to improve bearing performance. Similar situations may also be encountered when the lubricant becomes contaminated by dirt or metal particles. The Christensen model[8] does not account for these nonNewtonian situations. A question therefore naturally arises as to how the bearing performance would be modified in such situations, or especially when additives are used in lubricants. Microscopic effects, generated by micromotions of particles suspended in viscous fluids, drastically change the character of flow between narrow passages. One way of accounting for these microscopic effects is through the so-called theory of fluid microcontinua, wherein it is assumed that the microvolume elements consist of sets of structured particles, which not only contain mass, but can translate, rotate and deform. Micropolar fluid theory, advanced by Eringen[20], which ignores the deformation of the microelement, has been applied successfully by many authors[21-25] to study various problems in lubrication, where either the lubricant is assumed to be nonNewtonian or additives are used. The importance of this theory in the E H L parallel zone has been established by Prakash and Christensen [26]. They have noted that the additive-caused film thickness enhancement, as observed by several researchers in heavily loaded contacts, may have a possible explanation in this theory. Hence this theory is applied to Christensen's model to study the problem of surface roughness from a microcontinuum view point. For simplicity it is assumed that the moving surface is smooth whilst the stationary surface is rough, Fig. l(c).
h
= h n • h s
Fig. l(c). Geometry of rough journal bearing.
P. S1NHA and C. SINGH
622
The present study may have important consequences where the combined effects o f s u r f a c e r o u g h n e s s a n d a d d i t i v e s a r e o p e r a t i v e . It is a l s o f e l t t h a t t h e p r e s e n t t h e o r y m a y g i v e a b e t t e r i n s i g h t i n t o t h e r h e o l o g i c a l a b n o r m a l i t i e s so f r e q u e n t l y o b s e r v e d b y various researchers. MODIFIED REYNOLDS EQUATION FOR ROUGH SURFACES The one-dimensional Reynolds equation for a micropolar fluid for smooth surfaces has been derived by Prakash and Sinha[21] and the two-dimensional Reynolds equation by Shukla and lsa[24] and Tipei[25]. This equation in two dimensions, subject to the condition of constancy of viscosity coefficients across the film, is of the form 0 , x'+ 0 . Oh ~ t q I ~ { q , t =- 0--[
(1)
where qx and qy are fluxes along and perpendicular to the direction of motion of the surface (assumed along the x-axis) and have the forms Uh qx = 2
f ( N , a, h) Op 12/t Ox
(2)
and f ( N , a, h) Op 12tx 0y'
qY
(3)
where Mh f ( N , a, h) = h 3+ 12a2h - 6 N a b 2 coth ~ a
\~-~/
[
and N = \2--~--+-~] "
(4) (5)
U is the velocity of the moving surface, p is the hydrodynamic pressure, h is the film thickness of the lubricant, O. is the dynamic viscosity and ~. and 3' are new viscosity coefficients characterizing the micropolarity of the fluid. The geometry of the lubricant film can be considered to consist of two parts. The first part denotes the nominal (smooth) part of the geometry and can be considered to be a function of space and time coordinates. The second part of the film geometry, a randomly varying quantity, arises due to the surface roughness and is measured from the nominal level. Mathematically, h = hA(x, y, t) + h~(x, y, ~),
(6)
~: is a random variable which determines a definite roughness arrangement, from a large number of similarbut not identical-arrangements having the same statistical properties. Taking the expected value, equation (1) assumes the form ~x {E(q~)} +
OE(h) ± {E(q,)} = 0r ~'
(7)
where f~ E ( ) = J_~ ()f(tS)d8
(8)
and f(8) is the probability density distribution of the random variable & Using (8) it is readily seen [8], that O E ( h ) _ Oh,, at at '
(9)
where h, is the smooth part of the film thickness. To evaluate the average of the fluxes appearing in equation (7), subject to a particular, specific model of the roughness, i.e. either along or perpendicular to the sliding direction, two assumptions, similar to those used by Christensen[8], are made. (i) The pressure gradient in the roughness direction is assumed to be a stochastic variable with zero (or negligible) variance. (ii) The flux perpendicular to the roughness direction is assumed to be a stochastic variable with zero (or negligible) variance. Justification of these assumptions for a micropolar fluid can be made in a way similar to that of Christensen 18]. Henceforth a bar on a variable will represent the expected (or average) value of that quantity.
Lubrication of rough surfaces--a microcontinuum analysis
623
Longitudinal, one-dimensional roughness The roughness is assumed to run parallel to the sliding direction in the form of long narrow ridges and valleys. Thus the film geometry assumes the form
h = hn(x, y, t)+ h,(y, O.
(10)
Taking the average of all terms in equation (2) and adopting the first assumption the mean flux in the x-direction is given by
(11)
qx _ = ~U E(h)_l_~~ Eft(N, a, h))~xx.
This is so because aplax is a variable with zero variance and f(N, a, h) and aplox can be considered (approximately) to be stochastically independent quantities. In order, to get the mean flux in the y-direction, we divide equation (3) by f(N, a, h). Taking the expected value of both sides of the resulting equation and using the second assumption ap t]Y = - 12/'LE(~(N, la, h_~) Oy" 1
1
(12)
Substituting the expressions for fix and fly in equation (7) the modified Reynolds equation becomes
a F.
1
a~l
L v(N, a,
h)/
aE(h)+
#E(h)
#t "
(13)
j
Transverse, one-dimensional roughness In this model, the roughness striation is assumed to run perpendicular to the sliding direction in the form of long narrow ridges and furrows. The film thickness therefore assumes the following form
h = h,(x, y, t)+h,(x, O.
(14)
In a manner similar to that adopted earlier, the fluxes, along and perpendicular to the sliding direction, are E
h 05)
(16)
fly = - 1 - ~ Eft(N, ah)) ~yy,
and the modified Reynolds equation is -
a RE( 1
E
h
a~xx]+-~[E(f(N'a'h))~YY] = Ix ~ x ] ~ - - - ~ l
+12~
Ot
Uniform, isotropic roughness In this model the roughness is assumed to be uniformly distributed over the bearing surface with no preferred position or direction in the surface. The film thickness assumes the form
h = h,(x, y, t) + h,(x, y, O.
(18)
Adopting the same hypothesis as in [8] and mathematical reasoning given by Tonder [27], the fluxes and modified equations are given as follows. U
1
~913
fix = -~ E(h)-~-~ E(f(N, a, h))~-~,
(19)
fl, =-l-~y Eft(N, a, h))~,
(20)
624
P. SINHA and C. SINGH
and
O--x
~x +
E(f(N,a,h))
= 6p.U ~ - +
12~
(21)
at
Distribution of roughness heights The roughness distribution function which is generally used to evaluate the several expected values is, [8], f(hs)
35
2
= 3~-~(b - hs2)3, =0
elsewhere - b < h, < b
(22)
where b is one half the total range of the random film thickness variable and if tr is the standard deviation, b = -+ 3~. The polynomial distribution function (22) is an approximation to the Gaussian distribution. The reason for using such a polynomial distribution is that the Ganssian distribution always implies a finite probability of an asperity of a v e r y large size and thus violating the conditions for hydrodynamic operation. This type of approximation to the Gaussian distribution has been widely used in roughness problems[2, 3, 6, 7, 8, 9]. BEARING CHARACTERISTICS Once the modified Reynolds type equation is obtained, for a particular case, the evaluation of the bearing characteristics can be carried out in a way similar to that of the smooth bearing. (i) Oil ]low: When the roughness is longitudinal the mean flux along and across the direction of sliding is given by equations (11) and (12), respectively. For transverse roughness these values are governed by (15) and (16). Equations (19) and (20) account for these characteristics for the isotropic case. The total flow over the edge of the bearing can be found from the above mentioned equations by integrating in a usual way. (ii) Load capacity: The mean load carrying capacity is obtained by integrating the mean pressure over the bearing surface as (23)
= fA 1~dA (iii) Frictional drag. The expression for the shear stress on the moving surface is [21],
h ap, = 2 '~ ~
~U g(N, a, h)'
(24)
where
Nh
g(N, a, h) = h - 2Na tan h - 2a - "
(25)
For longitudinal roughness the mean shear stress is given by 1 a~ "r =~ E ( h ) - ~ + ~UE (g--~,l, h)). _
(26)
When the roughness is transverse,
h ,=~U
3E(f~)+E(~(N,a
'
-
2l
E(f~-~'J+2E{~f~]l
a' ~Ox"
(27)
The mean frictional drag is
f = fn ~rdA.
(28)
(iv) Coe~cient of friction. The mean coefficient of friction is defined by
~/ = f/f¢
(29)
Lubrication of rough s u r f a c e s i a microcontinuum analysis
625
MODIFIED REYNOLDS EQUATION APPLIED TO A ROUGH JOURNAL BEARING The modified Reynolds equation derived above will now be applied to the lubrication of a onedimensional rough journal bearing under half Sommerfeld boundary conditions. The configuration of the system is shown in Fig. l(c). Here the bearing is assumed to be rough and the journal to be smooth. The two cases of roughness, i.e. longitudinal and transverse, are treated separately.
Longitudinal, one-dimensional roughness Since x ffi RO, equation (13) assumes the form
d [ dO LEft(N, a,
~,. d~ ] = " UR dE(h) n ) j - ~ j ¢l.Lu -d-ff •
(30)
The smooth part of the film thickness is h~ = c(1 +E cos 0)
(31)
E(h) = h..
(32)
/5 = 0 at 0 = 0 and 0 = ~r.
(33)
and
Pressure distribution The boundary conditions for pressure are
The mean pressure distribution is given by
f o (hn - h~)dO /5(0) = 61~ UR )o Eft(N, a, h))'
(34)
where
(" =
h~dO
J0 E(ftn--~ h))
•,,o f , ,
dO
(35)
"
J0 E-- -ff Load capacity The mean load carrying capacity per unit length along and perpendicular to the line of centres are obtained by integrating the mean pressure around the bearing surface from 0 = 0 to 0 = ~r. Thus the mean load component normal to the line of centers per unit length is given by g'~/2= w sin 4~ =
/5(0) sin 0 R dO.
(36)
ff',~: = 6~UR 2 fo~ (hn - h~o)cos 0 dO Eft(N, a, h)) "
(37)
Substituting the value of ~(0) from (17) and integrating,
Similarly, the mean load component along the line of centres is given ..... 2 f~ (hn - h~) sin 0 dO 90 = • cos q~ = ol~o~ Jo Eft(N, a, h)) "
(38)
The resultant mean load capacity is given by
~' = (~'2/2-t- ~2)1/2.
(39)
Frictional drag The expression for mean shear stress (23) takes the form ~_E(h)d~ + 2R dO
(~1 \g(N,a.h)]"
(4O)
The mean frictional drag, L per unit length is given by E
1
R dO
(4.
or, using (34),
[lf'h.(h.2hDdO 1 r2" E 1 fffi61zUR[2Jo E f t ( N , a , h ) ) +61o ( ~ )
dO].
(42)
626
P. SINHA and C. SINGH
Coefficient of friction The coefficient of friction is obtained according to equation (29).
Transverse, one-dimensional roughness For this roughness the Reynolds equation is
dO|E[
"~[ "Ef -1 ~ d-° l= 6tl L V(N, a, h)/ J
L
I_ V(N, a,~/
(43)
"
In a way similar to that of the case of longitudinal roughness, the mean values of pressure distribution, load carrying capacity, frictional drag and coefficient of friction are, 0
h
1~(0) = 6Ix UR fo { E ( f ~ ) - h . , E ( f ~ ) } d O ,
(44)
~i,,,/:=6~UR fo { E ~ ~ ) - h . , E ( f ~ ) } c o s O d O ,
(45)
h '~°=6" URf0 {E(.:~)-hn' ~(:(N,la, hi)} sinOdO'
(46)
and
I : = 6 , UR'
E
h
fo°
dO h
2 (47)
where, ~E
h
(48)
The coefficient of friction is obtained as usual. For the one dimensional case the isotropic roughness will be equivalent to the longitudinal roughness.
Non-dimensional forms To obtain the numerical results the following non-dimensional scheme is adopted,
H = h./c
= 1 + e c o s 0, H o =
h.o/c, Ho, = hn,/c,
(49)
Hs = hJc, B = b/c, L = c/a, P = ~c2/6t~ UR, Ire = ~cZ/6tz UR 2 and F = fc/6t~ UR. T h e non-dimensional average values of bearing characteristics for longitudinal and transverse roughness are given as follows Longitudinal, one-dimensional roughness,
15(0) =
I: S,(N, L, H, B)' (H - H0)d0
f~ (H - Ho) cos 0 dO ff',,/z = if'sin 4, . . . . . . . . , J0 Sm(N,L, H, B)
f~(n-
no) sin o_do
if'0 = if'cos 6 = Jo
SI(N, L, H, B) '
I f " H ( H - Ho)dO
1 f2.
F=2,~ S~(N,L,H,B) ~6Jo G(N, L, H, B)dO, = ( W ~ 2 -t-
Wo2)1/2,
(50) (51) (52) (53) (54)
and
R _ P (c)m =~,
(55)
Lubrication of rough surfaces--a microcontinuum analysis
627
where fS
Ha =
H dO S,(N, L, H, B)
(56)
f0~ Sj(N, dO L, H, B) ' fs (n + H,)~(B: - H~) 3c o t h - ~ (H + H,) d H , s,(n, L , . , B) = n ~+ -~--~H + -12H U - - ~105N J_~
(57)
and (B 2- Hs2)3 dHs
35 f"
G(N, L, H, B)=3-~
J-e(H +
(58)
H,)-2--NL tanh~(H+ H,)"
For transverse, one-dimensional roughness,
Io'
{S3(N,L, H, B ) - HoiS2(N, L, H, B)}dO,
(59)
{S3(N, L, H, B) - Ho~S~(N, L, H, B)} cos 0 dO,
(60)
{S3(N,L, H, B) - HoIS2(N, L, H, B)} sin 0 dO,
(61)
15(0) = #./2 =
Io
#o =
P°= [~ fo" [S32(N' s--~, L,, L' H, H' B) HotS3(N,L, H, B)IdO I f2,, t + () J0 ~3S,(N, L,
.~S3Z(N, L, H, B)~dO] H, B) + G(N, L, H, B)- _ S2(N, L, H, B) J J'
(62)
where
f=Jo S2(N, L, H, B)dO
H0,
fo
(63)
S3(N, L, H, BRIO
and
35 fB
(H + H,/-2(B 2- H,2)3d H,
S,(N,L,H,B)=~j_B(H+H ,,t n 6N x3.,_12 ~ (H-+TH,)
.. + Hs)zcoth~
, i = 2, 3, 4. (64)
(H + Hs)
To determine the rate of change of bearing characteristics for a rough bearing as compared to the smooth theory the following bearing parameters are defined to study the micropolar effects. (i) Mean load ratio parameter. The mean load ratio parameter is defined by
wR--- (~)B=-----~
(65)
(i) Mean frictional drag parameter. The mean frictional drag ratio parameter is defined by Fa - F - (F).~o
(F).-o
(66)
(ffi) Mean coeDicient of friction parameter. The mean coefficient of friction parameter is defined by Cs = 12f- (/i/)B=0
(Pl).~o
(67)
In all the above parameters ( )e-o denotes the corresponding value for smooth surfaces.
RESULTS AND DISCUSSION (i) Dimensionless parameters. Three dimensionless parameters are important in connection with the present study, namely, N, L, and B, defined by equations (5) and (49). The parameter B(= b/c) is a characteristic of bearing geometry and arises due to the surface roughness.
628
P. SINHA and C. SINGH
It is obvious that the roughness effects would be accentuated when b ( B = b/c) is comparable to the bearing clearance c, i.e. when b is large and c is small. A detailed discussion and limitation of this parameter are given in [6-9]. Here it suffices to say that 0-< B < 1. B ~ 0 gives the case of smooth surfaces. The parameters N and L characterize the micropolarity of the lubricant and are of paramount importance in the microcontinuum analysis. Thermodynamic restrictions require that 0-< N 2 < 1/2 and 0 < L-< oo, [20, 25, 28]. These parameters have been discussed in great detail by Prakash and Sinha [21] and Prakash and Christensen[26]. Here therefore it suffices to say that the parameters N and L, may in some way, be linked to fluid properties-say the concentration and the molecular chain length of the additives, respectively. It is expected that the micropolar effects will be of importance either when the characteristic material length a ( L = c/a) is large, corresponding to a smaller value of L(say to a large size of additive molecule) or when the clearance width is small. The effects are also likely to be significant for larger values of N (which may signify a higher concentration of additive molecule in the lubricant). (ii) Limiting cases. T h e present study can be considered as a generalization of the works of Christensen[8] and Prakash and Sinha[21], because their results can be obtained from the present analysis by considering certain limiting cases. Christensen's results are obtained in the limit of L--* oo or N--* 0, as both of these cases correspond to the Newtonian results. The results of Prakash and Sinha are obtained in the limit B ~ 0 , since B--*0 signifies a smooth surface. (iii) Bearing characteristics. The analysis of surface roughness effects, from a microcontinuum view point, yields bearing characteristics which are functions, not only of the roughness parameter B, but also of the micropolar parameters N and L. The load ratio parameter WR, friction ratio parameter FR and the coefficient of friction ratio parameter CR have been plotted as a function of the roughness parameter B for various values of the coupling number N(e = 0.5, L = 30) in Figs. 2--4. It is seen that, akin to the Newtonian lubricants, the transverse roughness leads to a significant increase in load carrying capacity (Fig. 2) and frictional drag (Fig. 3) whereas the coefficient of friction decreases significantly (Fig. 4). The percentage rate of increase or decrease is higher for higher values of N. In the case of longitudinal roughness there is a slight decrease in load carrying capacity (Fig. 2) and a slight increase in frictional drag (Fig. 3), consequently the coefficient of friction increases for all values of N. Again it is seen that the rate of increase is further enhanced by large values of N. Figures 5-7 show the variation of the parameters Wa, FR and CR vs L for various values of N(~ = 0.5, B = 0.49) for longitudinal roughness. It is seen that the parameter WR is lower for higher values of N and intermediate values of L whereas FR is always higher. The reduction in • and the increase in P leads to a significant increase in coefficient of friction and the rate of decrease or increase is always higher for the micropolar fluid. However, the limits L-* 0 and L--* = give rise to the Newtonian values for these parameters. Figures 8-10 are the variation of the parameters WR, F~ and CR for transverse roughness. We and FR are positive showing that the load capacity and frictional drag are higher for rough hearings. However negative values of CR show the reduction in coefficient of friction. The rate of increase of load capacity and frictional drag is higher for micropolar fluids. Alternatively, the coefficient of friction parameter CR shows a typical trend. From Fig. 10 it can be concluded that for smaller values of L the rate of decrease of the coefficient of friction ( ~ ) is smaller for micropolar fluids. However for higher values of L the decrease is higher and then approaches the Newtonian rate (corresponding to L--* oo).
125
I
O0 50~
N=0 7 - ~ .
/
N=o
075
N=O 0 -
~
050
T Wn 025
O0
LONG IT
U
D
-
~
I
~
/
/
-0.25 / -0 500'0
j 01
012 B
N:0'0 / 013 :"
I 0 z,
04! 0
5
Fig. 2. Load parameter WR vs roughness parameter B for various values of coupling number N.
Lubrication of rough surfaces--a microcontinuum analysis
629
0 34 0.32
0.24 N=O .7 N=0.5 016
N:0.0~
008 00[ - - - " ~ ' - " - ~ L O N G I T U D I N A L -0 0/, -
Z0 0
01 0
8
0.2
03 ~
B
0.4.
5
-
Fig. 3. Frictional drag parameter FR vs roughness parameter B for various values of coupling number N.
o3oi 020
,
,
,
I N=05
010 ~
CR
y
-0.I0 -020
N=0 0 / / , , ~
-030
N=O?"
-O.Z.0
0~1
0j
2
0 ~3 B
04
04~ 3.5
•
Fig. 4. Coefficient of friction parameter CR vs roughness parameter B for various values of coupling number
MS VoL 24, No. 10--D
N.
630
P. SINHA and C. SINGH
-012
[
I
LONGITUDINAL
l
[
ROUGHNESS
=07 -0.116 E=05
B= 0.49
-0.112
T
WR -0.108
N=0.5
-0.104
N = 0.0
- 0.100
-0 0980
8
Newtonian
16 L
24
32
>
Fig. 5. Load parameter WR vs length ratio parameter L for various values of coupling number N ( B = 0.49, longitudinal roughness).
0.09
I
I
I
I
[
LONGITUDINAL
]
I
I
ROUGHNESS
N=0.7 E=0.50
0.07
T
%
0.05
003t
I
I
8
I
I
I
18
I
24
I
J
32
L Fig. 6. Frictional drag parameter FR versus length ratio parameter L for various values of coupling number N ( B = 0.49, longitudinal roughness).
Lubrication of rough surfaces--a microcontinuum analysis 0.220
i
I
I
631
i
i
i
i
f
I
I
24
I
I
32
0.200
T CR
0.180
/
N=O0
0.160
0.1500
I
~
i
16
I
L ) Fig. 7. Coefficient of friction parameter CR vs length ratio parameter L for various values of coupling number N(B = 0.49, longitudinal roughness). i
099
I
i
i
i
6=05 B
049
096
093
T WR
0 90 N=05
087
084 N= 0 . 0 0.81
i 12
J 24
L 36 L ~
NE WTONIAN
I 48
J 60
I 72
84
Fig. 8. Load parameter WR vs length ratio parameter L for various values of coupling number N(B = 0.49, transverse roughness).
CONCLUSIONS
The microcontinuum analysis of surface roughness in lubrication leads to the following conclusions (1) The qualitative trend of various bearing characteristics is similar to the Newtonian trend, for both types of surface roughness. (2) The rate of change of load capacity as well as the frictional drag is greater as compared with the Newtonian analysis.
632
P. SINHA and C. SINGH 037
T
I
~N=O.7
I
1
TRANSVERSE
I
ROUGHNESS
0 34
0.31 R
0.28
f
-N = 0.0 I
0.2 5
NEWT ONIAN I
12
I
24
I
36
I
48
L--
60
72
:'-
Fig. 9. Frictional drag parameter FR vs length ratio parameter L for various values of coupling number N(B = 0,49, transverse roughness). -0.3433
I
I
i
i
- 0l 3/.,i
-033
i
I
[
N=0.7
6=0.5 B = 0.49 N=0.5
-0.32
CR -0.30
TRANSVERSE ROUGHNESS
-0.29
_012833 [
I
12
i
24
..1/
6
I
48 L ~
l
60
I
72
~
8
|
96
Fig. 10. Coefficient of friction parameter Cs vs length ratio parameter L for various values of coupling number N ( B = 0.49, transverse roughness).
(3) Longitudinal roughness results in an increase in coefficient of friction and this rate of increase is enhanced by micropolarity of the fluid. Transverse roughness causes the coefficient of friction to be decreased. However, the rate of decrease is smaller for lower values of L and for higher values of L it is higher, as compared to those for Newtonian fluids. The results obtained are in conformity with various experimental evidences of increased effective viscosity in proximity of a solid surface. It can thus be concluded that roughness does play a part in the emergence of the so called rheological abnormalities.
Lubrication of rough surfaces--a microcontinuum analysis
633
Acknowledgements mThe authors sincerely acknowledge the helpful comments of the reviewers. Thanks are also due to Prof. W. Johnson for his suggestions in the revision of the paper. One of the authors (C.S.) sincerely acknowledges the financial grant awarded by the Council of Scientific and Industrial Research, New Delhi, India during the preparation of this manuscript. REFERENCES 1. R. A. BURTON,Effect of two dimensional, sinusoidal roughness on the load support characteristics of a lubricant film. J. Basic Engng. Trans. ASME, Series D 85, p. 258, (1963). 2. S. T. TZENG and E. SAIBEL, Surface roughness effect on slider bearing lubrication. ASLE. Trans. 10, p. 334 (1967). 3. S. T. Tz~Nc and E. SAIBEL,On the Effects of Surface Roughness in the Hydrodynamic Lubrication Theory of a Short Journal Bearing, Wear, 10, 179 (1967). 4. R. OSTVIKand H. CHmSTENSEN,Changes in surface topography with running-in. Proc. Instn. Mech. Engrs. 183 (Pt. 3P), p. 57 (1968-69). 5. D. F. MOOP,E, Drainage criteria for runway surface roughness, J. Roy. Aeron. Soc. 69, No 653, p. 337 (1965). 6. H. CHmSTENSENand K. TONDER,Tribology of Rough Surfaces: Stochastic Models for Hydrodynamic Lubrication. SINTEF Rep. 10/69-18 (1969). 7. H. CHRISTENSENand K. TOND~, Tribology of Rough Surfaces: Parametric Study and Comparison of Lubrication Models, SINTEF Rep. 22]69-18 (1969). 8. H. CHmSTENSEN,Stochastic models for hydrodynamic lubrication of rough surfaces. Proc. Instn. Mech. Engrs., 184 (Pt. 1) p. 1013. (1969-70). 9. H.G. ELROD,Thin-film lubricationtheory for Newtonian fiuids with surfaces possessing striated roughness or grooving. J. Lub. Tech., Trans. ASME. 95, p. 484 (1973). 10. A. JACKSONand A. CAMERON,An interferometric study of the EHL of rough surfaces, ASLE, Trans. 19, p. 50 (1975). 11. J C. HESlqIK~, The Depth of the Surface Zone of a liquid. Rev. Mod. Phys. 21, p. 322 (1949). 12. S. J. NEEDS, Boundary Film Investigations, Trans. ASME, 62, p. 331 (1940). 13. A. T. J. H^YWARDand J. D. ISDALE,The rheology of liquids very near to solid boundaries. Brit. J. Appl. Phys. 2, p. 251 (1969). 14. T. C. ASKWrrH,A. CAMERONand R. F. CROUCH,Chain length of additives in relation to lubricant in thin film and boundary lubrication, Proc. Roy. Soc. (London), Series A. 291, p. 500 (1966). 15. A. C ~ o N and R. GOHXR,Theoretical and experimental studies of the oil film in lubricated point contact. Proc. Roy. Soc. (London), Series A, 291, p. 520 (1966). 16. J. W. KA~EL, J. C. BELL, J. A. W~Lowrr and C. M. ALLEN,A Study of the Influence of Lubricants on High Speed Rolling Contact Bearing Performance, Battelle Memorial Institute ASD-TDR-61-643, Part VIII, (June 1968). 17. R. J. PARKERand J. W K~'NEL, Elastohydrodynamic Film Thickness Between Rolling Disks with a Synthetic Paraflinic Oil to 589 K, NASA TND-6411, (July 1971). 18. E. DRAUGLIS,A. A. LUCASand C. M. ALLEN,Smectic Model for Liquid Films on Solid Substrates, Spec. Disc. Far. Soc. 1, p. 251 (1971). 19. G. BOLLANI,Failure Criteria in Thin Film Lubrication With EP Additives. Wear. 36, p. 19 (1976). 20. A. C. ERINOEN,Theory of micropolar fluids. J. Math. Mech. 16, p. 1 (1966). 21. J. PP,AKASH and P. Srt~rlX, Lubrication theory for micropolar fluids and its application to a journal bearing, Int. J. Eng. Sci. 13, p. 217 (1975). 22. P. S[NH^, Effects of rigid particles in the Lubrication of Rolling Contact Bearings Considering Cavitation. Wear 44, p. 295 (1977). 23. M. S. KHADERand R. I. VACHON,Theoretical effects of solid particles in hydrostatic bearing lubricant, J. Lub. Tech., Trans. ASME 95, p. 104, (1973). 24. J. B. SHUKLAand M. ISA, Generalized Reynolds equation for micropolar lubricants and its application to optimum one-dimensionalslider bearings: Effects of solid particle additives in solution. J. Mech, Eng. Sci. 13, p. 217 (1975). 25. N. TIPEI,Lubrication with micropolar liquids and its application to short bearings. J. Lub. Tech., Trans. ASME. 101, p. 356 (1979). 26. J. PXXKASHand H. CHmST~NSEN,A Microcontinuum Theory for the Elastohydrodynamic Inlet Zone, J. Lub. Tech., Trans. ASME. 99, p. 24, (1977). 27. K. TONDER,Lubrication of Surfaces Having Area Distributed Isotropic Roughness. Joint ASME-ASLE Lubrication Conf. Boston (Oct. 1976). 28. P. SINH^ and C. SINOH, Theoretical effects of rigid panicle additives in non-cyclic squeeze films. TRANS. ASME, J. Lab. Tech. Paper No. 81-Lub-28. Also presented at the ASLE-ASME ?oint Lubrication Conf. New Orleans, LA, U.S.A. (5-7 Oct., 1981).
0020-7403/82/100619-15503.0010 © 1982 Pergamon Press Ltd.
LUBRICATION OF ROUGH S U R F A C E S - A MICROCONTINUUM ANALYSIS PRAWAL SINHA a n d CHANDAN SINGH Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur-208016, India SummarykNumerous papers have been devoted to the study of the influence of surface roughness on bearing systems. These studies, however, do not account for the non-Newtonian situations which arise so frequently in present day lubrication practice. One way of studying these non-Newtonian effects is through microcontinuum analysis, wherein the micromotions and deformation of fluid microcontinua are considered. These microscopic effects may be of great significance when the surfaces are rough. The micropolar fluid theory, in which the deformation of the microelement is ignored, is therefore applied to study the effects of surface roughness in lubrication. The problem considered is that of a rough journal bearing, where, for convenience, only the stationary part of the bearing is assumed to be rough. In line with Christensen's stochastic theory, the cases of longitudinal and transverse roughness are studied. The results obtained are in qualitative agreement with Christensen's results. However, it is shown that the effects of surface roughness are more pronounced, when analysed from a microcontinuum view point. NOTATION A surface area b half total range of random film thickness variable B b/c C bearing clearance CR coefficient of friction parameter E() expected (mean) value of () f frictional drag fO probability function f(N, a, h) defined by equation (4) F dimensionless frictional drag FR frictional drag parameter S(N, a, h) defined by equation (25) G(N, L, H, B) defined by equation (58) h h = h. + h,, film thickness for rough bearing h. smooth part of film thickness h, random part of film thickness h.o, h.~ defined by equations (35) and (48), respectively H, Ho, Ho. H, defined in equation (49) a characteristic material length, (-//4~) 112 L length ratio, c/a N coupling number (~,1(2/~+ 7,))"2 P hydrodynamic pressure, in general, a random variable P dimensionless pressure q . q~ fluxes in x and y directions, respectively R journal radius &(N, L, H, B) defined by equation (57) S~(N, L, H, B) defined by equation (64), i = 2, 3, 4 t time U tangential journal velocity Ww/2,W0 load capacities perpendicular and along the line of centers, respectively W resultant load capacity W,~2, W0, W dimensionless forms of w.12, w0 and w, respectively NR load ratio parameter X, y cartesian coordinates viscosity coefficients for micropolar fluids E eccentricity ratio 0 angular coordinate, x = RO dynamic viscosity coefficient of friction random variable I" shear stress A bar written above a variable denotes the mean or expected value of the corresponding variable. 619
620
P. SINHA and C. SINGH INTRODUCTION
The analysis of surface roughness effects on lubrication has attracted considerable attention and has become a subject of intensive study during the last two decades. This is due to their importance in the practical field, their physical significance and also their great theoretical interest. Most theoretical studies on bearing lubrication appearing in the literature make use of the assumption that the bearing surfaces are perfectly smooth. It has long been recognized that owing to machining limitations, this assumption is rather unrealistic, particularly in bearings, where the film thickness is small. In recent years a major way of analyzing the circumstances has been through the adoption of statistical methods. Two basic approaches have been developed, namely the deterministic and the stochastic approach. The starting point in each case has been the Reynolds equation. In the deterministic approach the surface roughness was accounted for by postulating a sine (cosine) wave or a series of sine (cosine) waves for the film thickness[I]. More recently Tzeng and Saibel[2, 3] have used stochastic concepts to study the surface roughness effects in slider bearings[2] and in short journal bearings[3]. The analysis was concerned with the problem of a two-dimensional slider bearing, with one-dimensional roughness in the direction transverse to the sliding direction. There is, however, experimental evidence [4] which often exhibits a one-dimensional type of roughness running in the direction of sliding, generally referred to as longitudinal roughness, Fig. l(a). Uni-directional roughness has also been observed and studied by Moore [5]. In line with this evidence, Christensen and Tonder[6, 7] and Christensen[8] presented a stochastic model for the hydrodynamic lubrication of rough surfaces. Christensen[8] considered two types of one-dimensional roughness, the longitudinal roughness, Fig. l(a), with its striations parallel to the sliding direction, and the transverse roughness, Fig. l(b), with its striations perpendicular to the sliding direction, and obtained a modified Reynolds equation by making two heuristic assumptions about the pressure gradient and flow rate. It was concluded that the longitudinal roughness results in a slight decrease in load-carrying capacity and an increase in frictional force thereby causing a significant increase in coefficient of friction. The effect of transverse roughness is, however, to improve the bearing characteristics. Later Elrod[9] improved Christensen's modified form of Reynolds equation. The theoretical results obtained by using Christensen's theory have recently been verified experimentally by Jackson and Cameron[10]. Scores of papers have appeared in the wake of these earlier studies on various aspects of the lubrication of rough surfaces. All of the works which have appeared to date show that the surface roughness may have significant effect when the minimum film thickness is comparable to the roughness mean height. The effect is much pronounced for lower values of film thickness. Inherent in these studies was a fundamental assumption that all the lubricants behaved as if they were Newtonian. y
Y j
.
j
......
x
~z
Fig. l(a). A schematic diagram of longitudinal roughness.
~/1111111/11111111
El- u-- F"
ih:x
I
I
/I/I//1111IIIIIIII/
l"
~Z
Fig. l(b). A schematic diagram of transverse roughness.
Lubrication of rough surfaces--a microcontinuum analysis
621
Considerable experimental evidence exists, Henniker [ l l], which however indicates that, fluids subjected to small clearances, such as those found in lubrication, may behave in a non-Newtonian way. Needs[12] also obtained results, which indicated a non-Newtonian behaviour in circular squeeze films. In a later review, Hayward and Isdale[13] concluded that rheological abnormalities in pure liquids do not extend to more than a few molecular diameters from a solid boundary. It was also indicated that the inherent surface roughness may be a cause for rigidity observed by the earlier workers. A remarkable result is that of Askwith et al. [14] and Cameron and Gohar[15] who report that there is a plastic layer of a rigidity sufficient to arrest completely the descent of a flat metal plate, at a depth of c a 1.9/~m (-~x 10-4) inch, even in pure cetane. Kannel et al.[16] observe nearly a 100% elasto-hydrodynamic film thickness enhancement when anti-wear additives are present as compared with the base fluid. The film enhancement has been reproducibly observed with synthetic paraffin base fluids as weU[17]. Similar results have been observed by Draugiis et al.[18] and
Bollani[19]. The overall conclusion was an increased effective viscosity in thin films. Such thin films in which the properties are no longer those of the bulk fluids, are of great technological significance, since most practical machine elements owe their performance to them, especially in the boundary or the mixed lubrication region. Non-Newtonian situations may also arise in bearings where additives are used. The use of additives these days is a common practice to improve bearing performance. Similar situations may also be encountered when the lubricant becomes contaminated by dirt or metal particles. The Christensen model[8] does not account for these nonNewtonian situations. A question therefore naturally arises as to how the bearing performance would be modified in such situations, or especially when additives are used in lubricants. Microscopic effects, generated by micromotions of particles suspended in viscous fluids, drastically change the character of flow between narrow passages. One way of accounting for these microscopic effects is through the so-called theory of fluid microcontinua, wherein it is assumed that the microvolume elements consist of sets of structured particles, which not only contain mass, but can translate, rotate and deform. Micropolar fluid theory, advanced by Eringen[20], which ignores the deformation of the microelement, has been applied successfully by many authors[21-25] to study various problems in lubrication, where either the lubricant is assumed to be nonNewtonian or additives are used. The importance of this theory in the E H L parallel zone has been established by Prakash and Christensen [26]. They have noted that the additive-caused film thickness enhancement, as observed by several researchers in heavily loaded contacts, may have a possible explanation in this theory. Hence this theory is applied to Christensen's model to study the problem of surface roughness from a microcontinuum view point. For simplicity it is assumed that the moving surface is smooth whilst the stationary surface is rough, Fig. l(c).
h
= h n • h s
Fig. l(c). Geometry of rough journal bearing.
P. S1NHA and C. SINGH
622
The present study may have important consequences where the combined effects o f s u r f a c e r o u g h n e s s a n d a d d i t i v e s a r e o p e r a t i v e . It is a l s o f e l t t h a t t h e p r e s e n t t h e o r y m a y g i v e a b e t t e r i n s i g h t i n t o t h e r h e o l o g i c a l a b n o r m a l i t i e s so f r e q u e n t l y o b s e r v e d b y various researchers. MODIFIED REYNOLDS EQUATION FOR ROUGH SURFACES The one-dimensional Reynolds equation for a micropolar fluid for smooth surfaces has been derived by Prakash and Sinha[21] and the two-dimensional Reynolds equation by Shukla and lsa[24] and Tipei[25]. This equation in two dimensions, subject to the condition of constancy of viscosity coefficients across the film, is of the form 0 , x'+ 0 . Oh ~ t q I ~ { q , t =- 0--[
(1)
where qx and qy are fluxes along and perpendicular to the direction of motion of the surface (assumed along the x-axis) and have the forms Uh qx = 2
f ( N , a, h) Op 12/t Ox
(2)
and f ( N , a, h) Op 12tx 0y'
qY
(3)
where Mh f ( N , a, h) = h 3+ 12a2h - 6 N a b 2 coth ~ a
\~-~/
[
and N = \2--~--+-~] "
(4) (5)
U is the velocity of the moving surface, p is the hydrodynamic pressure, h is the film thickness of the lubricant, O. is the dynamic viscosity and ~. and 3' are new viscosity coefficients characterizing the micropolarity of the fluid. The geometry of the lubricant film can be considered to consist of two parts. The first part denotes the nominal (smooth) part of the geometry and can be considered to be a function of space and time coordinates. The second part of the film geometry, a randomly varying quantity, arises due to the surface roughness and is measured from the nominal level. Mathematically, h = hA(x, y, t) + h~(x, y, ~),
(6)
~: is a random variable which determines a definite roughness arrangement, from a large number of similarbut not identical-arrangements having the same statistical properties. Taking the expected value, equation (1) assumes the form ~x {E(q~)} +
OE(h) ± {E(q,)} = 0r ~'
(7)
where f~ E ( ) = J_~ ()f(tS)d8
(8)
and f(8) is the probability density distribution of the random variable & Using (8) it is readily seen [8], that O E ( h ) _ Oh,, at at '
(9)
where h, is the smooth part of the film thickness. To evaluate the average of the fluxes appearing in equation (7), subject to a particular, specific model of the roughness, i.e. either along or perpendicular to the sliding direction, two assumptions, similar to those used by Christensen[8], are made. (i) The pressure gradient in the roughness direction is assumed to be a stochastic variable with zero (or negligible) variance. (ii) The flux perpendicular to the roughness direction is assumed to be a stochastic variable with zero (or negligible) variance. Justification of these assumptions for a micropolar fluid can be made in a way similar to that of Christensen 18]. Henceforth a bar on a variable will represent the expected (or average) value of that quantity.
Lubrication of rough surfaces--a microcontinuum analysis
623
Longitudinal, one-dimensional roughness The roughness is assumed to run parallel to the sliding direction in the form of long narrow ridges and valleys. Thus the film geometry assumes the form
h = hn(x, y, t)+ h,(y, O.
(10)
Taking the average of all terms in equation (2) and adopting the first assumption the mean flux in the x-direction is given by
(11)
qx _ = ~U E(h)_l_~~ Eft(N, a, h))~xx.
This is so because aplax is a variable with zero variance and f(N, a, h) and aplox can be considered (approximately) to be stochastically independent quantities. In order, to get the mean flux in the y-direction, we divide equation (3) by f(N, a, h). Taking the expected value of both sides of the resulting equation and using the second assumption ap t]Y = - 12/'LE(~(N, la, h_~) Oy" 1
1
(12)
Substituting the expressions for fix and fly in equation (7) the modified Reynolds equation becomes
a F.
1
a~l
L v(N, a,
h)/
aE(h)+
#E(h)
#t "
(13)
j
Transverse, one-dimensional roughness In this model, the roughness striation is assumed to run perpendicular to the sliding direction in the form of long narrow ridges and furrows. The film thickness therefore assumes the following form
h = h,(x, y, t)+h,(x, O.
(14)
In a manner similar to that adopted earlier, the fluxes, along and perpendicular to the sliding direction, are E
h 05)
(16)
fly = - 1 - ~ Eft(N, ah)) ~yy,
and the modified Reynolds equation is -
a RE( 1
E
h
a~xx]+-~[E(f(N'a'h))~YY] = Ix ~ x ] ~ - - - ~ l
+12~
Ot
Uniform, isotropic roughness In this model the roughness is assumed to be uniformly distributed over the bearing surface with no preferred position or direction in the surface. The film thickness assumes the form
h = h,(x, y, t) + h,(x, y, O.
(18)
Adopting the same hypothesis as in [8] and mathematical reasoning given by Tonder [27], the fluxes and modified equations are given as follows. U
1
~913
fix = -~ E(h)-~-~ E(f(N, a, h))~-~,
(19)
fl, =-l-~y Eft(N, a, h))~,
(20)
624
P. SINHA and C. SINGH
and
O--x
~x +
E(f(N,a,h))
= 6p.U ~ - +
12~
(21)
at
Distribution of roughness heights The roughness distribution function which is generally used to evaluate the several expected values is, [8], f(hs)
35
2
= 3~-~(b - hs2)3, =0
elsewhere - b < h, < b
(22)
where b is one half the total range of the random film thickness variable and if tr is the standard deviation, b = -+ 3~. The polynomial distribution function (22) is an approximation to the Gaussian distribution. The reason for using such a polynomial distribution is that the Ganssian distribution always implies a finite probability of an asperity of a v e r y large size and thus violating the conditions for hydrodynamic operation. This type of approximation to the Gaussian distribution has been widely used in roughness problems[2, 3, 6, 7, 8, 9]. BEARING CHARACTERISTICS Once the modified Reynolds type equation is obtained, for a particular case, the evaluation of the bearing characteristics can be carried out in a way similar to that of the smooth bearing. (i) Oil ]low: When the roughness is longitudinal the mean flux along and across the direction of sliding is given by equations (11) and (12), respectively. For transverse roughness these values are governed by (15) and (16). Equations (19) and (20) account for these characteristics for the isotropic case. The total flow over the edge of the bearing can be found from the above mentioned equations by integrating in a usual way. (ii) Load capacity: The mean load carrying capacity is obtained by integrating the mean pressure over the bearing surface as (23)
= fA 1~dA (iii) Frictional drag. The expression for the shear stress on the moving surface is [21],
h ap, = 2 '~ ~
~U g(N, a, h)'
(24)
where
Nh
g(N, a, h) = h - 2Na tan h - 2a - "
(25)
For longitudinal roughness the mean shear stress is given by 1 a~ "r =~ E ( h ) - ~ + ~UE (g--~,l, h)). _
(26)
When the roughness is transverse,
h ,=~U
3E(f~)+E(~(N,a
'
-
2l
E(f~-~'J+2E{~f~]l
a' ~Ox"
(27)
The mean frictional drag is
f = fn ~rdA.
(28)
(iv) Coe~cient of friction. The mean coefficient of friction is defined by
~/ = f/f¢
(29)
Lubrication of rough s u r f a c e s i a microcontinuum analysis
625
MODIFIED REYNOLDS EQUATION APPLIED TO A ROUGH JOURNAL BEARING The modified Reynolds equation derived above will now be applied to the lubrication of a onedimensional rough journal bearing under half Sommerfeld boundary conditions. The configuration of the system is shown in Fig. l(c). Here the bearing is assumed to be rough and the journal to be smooth. The two cases of roughness, i.e. longitudinal and transverse, are treated separately.
Longitudinal, one-dimensional roughness Since x ffi RO, equation (13) assumes the form
d [ dO LEft(N, a,
~,. d~ ] = " UR dE(h) n ) j - ~ j ¢l.Lu -d-ff •
(30)
The smooth part of the film thickness is h~ = c(1 +E cos 0)
(31)
E(h) = h..
(32)
/5 = 0 at 0 = 0 and 0 = ~r.
(33)
and
Pressure distribution The boundary conditions for pressure are
The mean pressure distribution is given by
f o (hn - h~)dO /5(0) = 61~ UR )o Eft(N, a, h))'
(34)
where
(" =
h~dO
J0 E(ftn--~ h))
•,,o f , ,
dO
(35)
"
J0 E-- -ff Load capacity The mean load carrying capacity per unit length along and perpendicular to the line of centres are obtained by integrating the mean pressure around the bearing surface from 0 = 0 to 0 = ~r. Thus the mean load component normal to the line of centers per unit length is given by g'~/2= w sin 4~ =
/5(0) sin 0 R dO.
(36)
ff',~: = 6~UR 2 fo~ (hn - h~o)cos 0 dO Eft(N, a, h)) "
(37)
Substituting the value of ~(0) from (17) and integrating,
Similarly, the mean load component along the line of centres is given ..... 2 f~ (hn - h~) sin 0 dO 90 = • cos q~ = ol~o~ Jo Eft(N, a, h)) "
(38)
The resultant mean load capacity is given by
~' = (~'2/2-t- ~2)1/2.
(39)
Frictional drag The expression for mean shear stress (23) takes the form ~_E(h)d~ + 2R dO
(~1 \g(N,a.h)]"
(4O)
The mean frictional drag, L per unit length is given by E
1
R dO
(4.
or, using (34),
[lf'h.(h.2hDdO 1 r2" E 1 fffi61zUR[2Jo E f t ( N , a , h ) ) +61o ( ~ )
dO].
(42)
626
P. SINHA and C. SINGH
Coefficient of friction The coefficient of friction is obtained according to equation (29).
Transverse, one-dimensional roughness For this roughness the Reynolds equation is
dO|E[
"~[ "Ef -1 ~ d-° l= 6tl L V(N, a, h)/ J
L
I_ V(N, a,~/
(43)
"
In a way similar to that of the case of longitudinal roughness, the mean values of pressure distribution, load carrying capacity, frictional drag and coefficient of friction are, 0
h
1~(0) = 6Ix UR fo { E ( f ~ ) - h . , E ( f ~ ) } d O ,
(44)
~i,,,/:=6~UR fo { E ~ ~ ) - h . , E ( f ~ ) } c o s O d O ,
(45)
h '~°=6" URf0 {E(.:~)-hn' ~(:(N,la, hi)} sinOdO'
(46)
and
I : = 6 , UR'
E
h
fo°
dO h
2 (47)
where, ~E
h
(48)
The coefficient of friction is obtained as usual. For the one dimensional case the isotropic roughness will be equivalent to the longitudinal roughness.
Non-dimensional forms To obtain the numerical results the following non-dimensional scheme is adopted,
H = h./c
= 1 + e c o s 0, H o =
h.o/c, Ho, = hn,/c,
(49)
Hs = hJc, B = b/c, L = c/a, P = ~c2/6t~ UR, Ire = ~cZ/6tz UR 2 and F = fc/6t~ UR. T h e non-dimensional average values of bearing characteristics for longitudinal and transverse roughness are given as follows Longitudinal, one-dimensional roughness,
15(0) =
I: S,(N, L, H, B)' (H - H0)d0
f~ (H - Ho) cos 0 dO ff',,/z = if'sin 4, . . . . . . . . , J0 Sm(N,L, H, B)
f~(n-
no) sin o_do
if'0 = if'cos 6 = Jo
SI(N, L, H, B) '
I f " H ( H - Ho)dO
1 f2.
F=2,~ S~(N,L,H,B) ~6Jo G(N, L, H, B)dO, = ( W ~ 2 -t-
Wo2)1/2,
(50) (51) (52) (53) (54)
and
R _ P (c)m =~,
(55)
Lubrication of rough surfaces--a microcontinuum analysis
627
where fS
Ha =
H dO S,(N, L, H, B)
(56)
f0~ Sj(N, dO L, H, B) ' fs (n + H,)~(B: - H~) 3c o t h - ~ (H + H,) d H , s,(n, L , . , B) = n ~+ -~--~H + -12H U - - ~105N J_~
(57)
and (B 2- Hs2)3 dHs
35 f"
G(N, L, H, B)=3-~
J-e(H +
(58)
H,)-2--NL tanh~(H+ H,)"
For transverse, one-dimensional roughness,
Io'
{S3(N,L, H, B ) - HoiS2(N, L, H, B)}dO,
(59)
{S3(N, L, H, B) - Ho~S~(N, L, H, B)} cos 0 dO,
(60)
{S3(N,L, H, B) - HoIS2(N, L, H, B)} sin 0 dO,
(61)
15(0) = #./2 =
Io
#o =
P°= [~ fo" [S32(N' s--~, L,, L' H, H' B) HotS3(N,L, H, B)IdO I f2,, t + () J0 ~3S,(N, L,
.~S3Z(N, L, H, B)~dO] H, B) + G(N, L, H, B)- _ S2(N, L, H, B) J J'
(62)
where
f=Jo S2(N, L, H, B)dO
H0,
fo
(63)
S3(N, L, H, BRIO
and
35 fB
(H + H,/-2(B 2- H,2)3d H,
S,(N,L,H,B)=~j_B(H+H ,,t n 6N x3.,_12 ~ (H-+TH,)
.. + Hs)zcoth~
, i = 2, 3, 4. (64)
(H + Hs)
To determine the rate of change of bearing characteristics for a rough bearing as compared to the smooth theory the following bearing parameters are defined to study the micropolar effects. (i) Mean load ratio parameter. The mean load ratio parameter is defined by
wR--- (~)B=-----~
(65)
(i) Mean frictional drag parameter. The mean frictional drag ratio parameter is defined by Fa - F - (F).~o
(F).-o
(66)
(ffi) Mean coeDicient of friction parameter. The mean coefficient of friction parameter is defined by Cs = 12f- (/i/)B=0
(Pl).~o
(67)
In all the above parameters ( )e-o denotes the corresponding value for smooth surfaces.
RESULTS AND DISCUSSION (i) Dimensionless parameters. Three dimensionless parameters are important in connection with the present study, namely, N, L, and B, defined by equations (5) and (49). The parameter B(= b/c) is a characteristic of bearing geometry and arises due to the surface roughness.
628
P. SINHA and C. SINGH
It is obvious that the roughness effects would be accentuated when b ( B = b/c) is comparable to the bearing clearance c, i.e. when b is large and c is small. A detailed discussion and limitation of this parameter are given in [6-9]. Here it suffices to say that 0-< B < 1. B ~ 0 gives the case of smooth surfaces. The parameters N and L characterize the micropolarity of the lubricant and are of paramount importance in the microcontinuum analysis. Thermodynamic restrictions require that 0-< N 2 < 1/2 and 0 < L-< oo, [20, 25, 28]. These parameters have been discussed in great detail by Prakash and Sinha [21] and Prakash and Christensen[26]. Here therefore it suffices to say that the parameters N and L, may in some way, be linked to fluid properties-say the concentration and the molecular chain length of the additives, respectively. It is expected that the micropolar effects will be of importance either when the characteristic material length a ( L = c/a) is large, corresponding to a smaller value of L(say to a large size of additive molecule) or when the clearance width is small. The effects are also likely to be significant for larger values of N (which may signify a higher concentration of additive molecule in the lubricant). (ii) Limiting cases. T h e present study can be considered as a generalization of the works of Christensen[8] and Prakash and Sinha[21], because their results can be obtained from the present analysis by considering certain limiting cases. Christensen's results are obtained in the limit of L--* oo or N--* 0, as both of these cases correspond to the Newtonian results. The results of Prakash and Sinha are obtained in the limit B ~ 0 , since B--*0 signifies a smooth surface. (iii) Bearing characteristics. The analysis of surface roughness effects, from a microcontinuum view point, yields bearing characteristics which are functions, not only of the roughness parameter B, but also of the micropolar parameters N and L. The load ratio parameter WR, friction ratio parameter FR and the coefficient of friction ratio parameter CR have been plotted as a function of the roughness parameter B for various values of the coupling number N(e = 0.5, L = 30) in Figs. 2--4. It is seen that, akin to the Newtonian lubricants, the transverse roughness leads to a significant increase in load carrying capacity (Fig. 2) and frictional drag (Fig. 3) whereas the coefficient of friction decreases significantly (Fig. 4). The percentage rate of increase or decrease is higher for higher values of N. In the case of longitudinal roughness there is a slight decrease in load carrying capacity (Fig. 2) and a slight increase in frictional drag (Fig. 3), consequently the coefficient of friction increases for all values of N. Again it is seen that the rate of increase is further enhanced by large values of N. Figures 5-7 show the variation of the parameters Wa, FR and CR vs L for various values of N(~ = 0.5, B = 0.49) for longitudinal roughness. It is seen that the parameter WR is lower for higher values of N and intermediate values of L whereas FR is always higher. The reduction in • and the increase in P leads to a significant increase in coefficient of friction and the rate of decrease or increase is always higher for the micropolar fluid. However, the limits L-* 0 and L--* = give rise to the Newtonian values for these parameters. Figures 8-10 are the variation of the parameters WR, F~ and CR for transverse roughness. We and FR are positive showing that the load capacity and frictional drag are higher for rough hearings. However negative values of CR show the reduction in coefficient of friction. The rate of increase of load capacity and frictional drag is higher for micropolar fluids. Alternatively, the coefficient of friction parameter CR shows a typical trend. From Fig. 10 it can be concluded that for smaller values of L the rate of decrease of the coefficient of friction ( ~ ) is smaller for micropolar fluids. However for higher values of L the decrease is higher and then approaches the Newtonian rate (corresponding to L--* oo).
125
I
O0 50~
N=0 7 - ~ .
/
N=o
075
N=O 0 -
~
050
T Wn 025
O0
LONG IT
U
D
-
~
I
~
/
/
-0.25 / -0 500'0
j 01
012 B
N:0'0 / 013 :"
I 0 z,
04! 0
5
Fig. 2. Load parameter WR vs roughness parameter B for various values of coupling number N.
Lubrication of rough surfaces--a microcontinuum analysis
629
0 34 0.32
0.24 N=O .7 N=0.5 016
N:0.0~
008 00[ - - - " ~ ' - " - ~ L O N G I T U D I N A L -0 0/, -
Z0 0
01 0
8
0.2
03 ~
B
0.4.
5
-
Fig. 3. Frictional drag parameter FR vs roughness parameter B for various values of coupling number N.
o3oi 020
,
,
,
I N=05
010 ~
CR
y
-0.I0 -020
N=0 0 / / , , ~
-030
N=O?"
-O.Z.0
0~1
0j
2
0 ~3 B
04
04~ 3.5
•
Fig. 4. Coefficient of friction parameter CR vs roughness parameter B for various values of coupling number
MS VoL 24, No. 10--D
N.
630
P. SINHA and C. SINGH
-012
[
I
LONGITUDINAL
l
[
ROUGHNESS
=07 -0.116 E=05
B= 0.49
-0.112
T
WR -0.108
N=0.5
-0.104
N = 0.0
- 0.100
-0 0980
8
Newtonian
16 L
24
32
>
Fig. 5. Load parameter WR vs length ratio parameter L for various values of coupling number N ( B = 0.49, longitudinal roughness).
0.09
I
I
I
I
[
LONGITUDINAL
]
I
I
ROUGHNESS
N=0.7 E=0.50
0.07
T
%
0.05
003t
I
I
8
I
I
I
18
I
24
I
J
32
L Fig. 6. Frictional drag parameter FR versus length ratio parameter L for various values of coupling number N ( B = 0.49, longitudinal roughness).
Lubrication of rough surfaces--a microcontinuum analysis 0.220
i
I
I
631
i
i
i
i
f
I
I
24
I
I
32
0.200
T CR
0.180
/
N=O0
0.160
0.1500
I
~
i
16
I
L ) Fig. 7. Coefficient of friction parameter CR vs length ratio parameter L for various values of coupling number N(B = 0.49, longitudinal roughness). i
099
I
i
i
i
6=05 B
049
096
093
T WR
0 90 N=05
087
084 N= 0 . 0 0.81
i 12
J 24
L 36 L ~
NE WTONIAN
I 48
J 60
I 72
84
Fig. 8. Load parameter WR vs length ratio parameter L for various values of coupling number N(B = 0.49, transverse roughness).
CONCLUSIONS
The microcontinuum analysis of surface roughness in lubrication leads to the following conclusions (1) The qualitative trend of various bearing characteristics is similar to the Newtonian trend, for both types of surface roughness. (2) The rate of change of load capacity as well as the frictional drag is greater as compared with the Newtonian analysis.
632
P. SINHA and C. SINGH 037
T
I
~N=O.7
I
1
TRANSVERSE
I
ROUGHNESS
0 34
0.31 R
0.28
f
-N = 0.0 I
0.2 5
NEWT ONIAN I
12
I
24
I
36
I
48
L--
60
72
:'-
Fig. 9. Frictional drag parameter FR vs length ratio parameter L for various values of coupling number N(B = 0,49, transverse roughness). -0.3433
I
I
i
i
- 0l 3/.,i
-033
i
I
[
N=0.7
6=0.5 B = 0.49 N=0.5
-0.32
CR -0.30
TRANSVERSE ROUGHNESS
-0.29
_012833 [
I
12
i
24
..1/
6
I
48 L ~
l
60
I
72
~
8
|
96
Fig. 10. Coefficient of friction parameter Cs vs length ratio parameter L for various values of coupling number N ( B = 0.49, transverse roughness).
(3) Longitudinal roughness results in an increase in coefficient of friction and this rate of increase is enhanced by micropolarity of the fluid. Transverse roughness causes the coefficient of friction to be decreased. However, the rate of decrease is smaller for lower values of L and for higher values of L it is higher, as compared to those for Newtonian fluids. The results obtained are in conformity with various experimental evidences of increased effective viscosity in proximity of a solid surface. It can thus be concluded that roughness does play a part in the emergence of the so called rheological abnormalities.
Lubrication of rough surfaces--a microcontinuum analysis
633
Acknowledgements mThe authors sincerely acknowledge the helpful comments of the reviewers. Thanks are also due to Prof. W. Johnson for his suggestions in the revision of the paper. One of the authors (C.S.) sincerely acknowledges the financial grant awarded by the Council of Scientific and Industrial Research, New Delhi, India during the preparation of this manuscript. REFERENCES 1. R. A. BURTON,Effect of two dimensional, sinusoidal roughness on the load support characteristics of a lubricant film. J. Basic Engng. Trans. ASME, Series D 85, p. 258, (1963). 2. S. T. TZENG and E. SAIBEL, Surface roughness effect on slider bearing lubrication. ASLE. Trans. 10, p. 334 (1967). 3. S. T. Tz~Nc and E. SAIBEL,On the Effects of Surface Roughness in the Hydrodynamic Lubrication Theory of a Short Journal Bearing, Wear, 10, 179 (1967). 4. R. OSTVIKand H. CHmSTENSEN,Changes in surface topography with running-in. Proc. Instn. Mech. Engrs. 183 (Pt. 3P), p. 57 (1968-69). 5. D. F. MOOP,E, Drainage criteria for runway surface roughness, J. Roy. Aeron. Soc. 69, No 653, p. 337 (1965). 6. H. CHmSTENSENand K. TONDER,Tribology of Rough Surfaces: Stochastic Models for Hydrodynamic Lubrication. SINTEF Rep. 10/69-18 (1969). 7. H. CHRISTENSENand K. TOND~, Tribology of Rough Surfaces: Parametric Study and Comparison of Lubrication Models, SINTEF Rep. 22]69-18 (1969). 8. H. CHmSTENSEN,Stochastic models for hydrodynamic lubrication of rough surfaces. Proc. Instn. Mech. Engrs., 184 (Pt. 1) p. 1013. (1969-70). 9. H.G. ELROD,Thin-film lubricationtheory for Newtonian fiuids with surfaces possessing striated roughness or grooving. J. Lub. Tech., Trans. ASME. 95, p. 484 (1973). 10. A. JACKSONand A. CAMERON,An interferometric study of the EHL of rough surfaces, ASLE, Trans. 19, p. 50 (1975). 11. J C. HESlqIK~, The Depth of the Surface Zone of a liquid. Rev. Mod. Phys. 21, p. 322 (1949). 12. S. J. NEEDS, Boundary Film Investigations, Trans. ASME, 62, p. 331 (1940). 13. A. T. J. H^YWARDand J. D. ISDALE,The rheology of liquids very near to solid boundaries. Brit. J. Appl. Phys. 2, p. 251 (1969). 14. T. C. ASKWrrH,A. CAMERONand R. F. CROUCH,Chain length of additives in relation to lubricant in thin film and boundary lubrication, Proc. Roy. Soc. (London), Series A. 291, p. 500 (1966). 15. A. C ~ o N and R. GOHXR,Theoretical and experimental studies of the oil film in lubricated point contact. Proc. Roy. Soc. (London), Series A, 291, p. 520 (1966). 16. J. W. KA~EL, J. C. BELL, J. A. W~Lowrr and C. M. ALLEN,A Study of the Influence of Lubricants on High Speed Rolling Contact Bearing Performance, Battelle Memorial Institute ASD-TDR-61-643, Part VIII, (June 1968). 17. R. J. PARKERand J. W K~'NEL, Elastohydrodynamic Film Thickness Between Rolling Disks with a Synthetic Paraflinic Oil to 589 K, NASA TND-6411, (July 1971). 18. E. DRAUGLIS,A. A. LUCASand C. M. ALLEN,Smectic Model for Liquid Films on Solid Substrates, Spec. Disc. Far. Soc. 1, p. 251 (1971). 19. G. BOLLANI,Failure Criteria in Thin Film Lubrication With EP Additives. Wear. 36, p. 19 (1976). 20. A. C. ERINOEN,Theory of micropolar fluids. J. Math. Mech. 16, p. 1 (1966). 21. J. PP,AKASH and P. Srt~rlX, Lubrication theory for micropolar fluids and its application to a journal bearing, Int. J. Eng. Sci. 13, p. 217 (1975). 22. P. S[NH^, Effects of rigid particles in the Lubrication of Rolling Contact Bearings Considering Cavitation. Wear 44, p. 295 (1977). 23. M. S. KHADERand R. I. VACHON,Theoretical effects of solid particles in hydrostatic bearing lubricant, J. Lub. Tech., Trans. ASME 95, p. 104, (1973). 24. J. B. SHUKLAand M. ISA, Generalized Reynolds equation for micropolar lubricants and its application to optimum one-dimensionalslider bearings: Effects of solid particle additives in solution. J. Mech, Eng. Sci. 13, p. 217 (1975). 25. N. TIPEI,Lubrication with micropolar liquids and its application to short bearings. J. Lub. Tech., Trans. ASME. 101, p. 356 (1979). 26. J. PXXKASHand H. CHmST~NSEN,A Microcontinuum Theory for the Elastohydrodynamic Inlet Zone, J. Lub. Tech., Trans. ASME. 99, p. 24, (1977). 27. K. TONDER,Lubrication of Surfaces Having Area Distributed Isotropic Roughness. Joint ASME-ASLE Lubrication Conf. Boston (Oct. 1976). 28. P. SINH^ and C. SINOH, Theoretical effects of rigid panicle additives in non-cyclic squeeze films. TRANS. ASME, J. Lab. Tech. Paper No. 81-Lub-28. Also presented at the ASLE-ASME ?oint Lubrication Conf. New Orleans, LA, U.S.A. (5-7 Oct., 1981).