Int. J. Engng Sci. Vol. 23. No.2. pp. 235-240. 1985Printed in the U.S.A.

0020-7225/85 $3.00 + .00© 1985 Pergamon Press Ltd.

STUDIES IN CONVERGENCE OF STOCHASTICMODELS IN ROUGH-SURFACE HYDRODYNAMIC

LUBRICATION THEORY

S. R. WU and J. T. ODENTICOM,The Universityof Texasat Austin,Austin.TX 78712, U.S.A.

Abstract-The stochasticReynoldsequation for hydrodynamiclubricationwith random homo-geneousroughnessof the lubricatedsurfaceisstudiedusingseriesexpansions.In the case in whichthe roughness function b E CO(fl), we show convergenceof the series for pressure and itsexpectationin the SobolevspaceH'(rl), whereasin the case in which0 E C1(fl), the seriesconvergein C2(11) providedVbis uniformlybounded.

1. INTRODUCTIONSINCE Tzeng and Saibel [I, 2] introduced stochastic methods for the analysis of roughbearings and Christensen [3] developed the averaged Reynolds equations, many papershave been published on stochastic procedures dealing with hydrodynamic lubricationproblems with bearings having rough surfaces. The validity of various averaging techniquesfor the associated mean Reynolds equations has been the source of much study anddebate.

Rhode and Whicker [4] assumed the film thickness H to be of the form H = h + EO

(0 is the roughness) and expanded the pressure into a series P = L Enpn, E > O. Then-O

a..

expectation (P) = L En(Pn) of the pressure is likewise an expansion, as well as then-O

inverse operator of the governing differential operator L. These authors were able toprove O(E2

) accuracy in (P) when using Po as the solution corresponding to smoothbearing. Phan-Thien [5, 6] used an iterative analysis for the same problem and alsoobtained O(E2) accuracy in p. In both of these papers, methods are advocated whichexploit the absence of E term in (P) and which make use of the assumption (0) = O. Inthe present note, we shall prove that the pressure p and its expectation (P) converge inthe H'(fl) Sobolev norm provided ° E CO(Q). Moreover, we establish that if ° E C'(Q),and '10 is uniformly bounded, then one can prove C2 convergence of P and of (P).

2. EXPANSIONSLet us consider the classical situation of flow of an isothermal incompressible

Newtonian lubricant through a rigid rough bearing. For so-called Reynolds roughness,the governing equation is still the Reynolds equation (see Fig. I)

LP-= -'1. (H3'1p) = -6µ a(UH)}axPlelo = 0

(I)

where fl is a smooth open domain in 712, U is the rolling velocity of the bearing in the xdirection, and H is the film thickness, given by

H = h + EO, E> O. (2)

Here h(x, y) is the mean film thickness coincident with the smooth bearing geometry ando(x, y) is the normalized roughness, a random process with zero expectation, (0) = O.

235

236 S. R. WU and J. T. ODEN

z

Fig. I. Rough bearing.

x

Let us assume that the lubricant pressure p can be represented by the expansion

P = L Enpnn=O

(3)

where E is an arbitrary positive number. Substituting eqn (3) into eqn (1) results in theperturbed operators,

aUhLoPo = -6µ- =10ax

auoLoP, = -6µ - - LIPo = ItaxLoP2 = -L2PO - LIP. =hLoPn = -L3Pn-3 - L2Pn-2 - L1Pn-1 = In,Pilou = 0

n~3

(4)

where the governing operator L in eqn (I) is now given by the sum,

whereLo = -'1- (h3'1)

L, = -3'1· (1120'1)

L2 = -3'1· (ha2'1)

L3 = -'1' (03'1). (5)

3. CASE I: 0 E CO(fll

Since the roughness 0 may be only continuous but not necessarily differentiable, weare led naturally to the need for formulating a variational or weak statement of thestochastic lubrication problem. We first recall the so-called generalized Lax-Milgramtheorem (cf. [7]): Let H be a Hilbert space and let B: H X H -+ 7l be a bilinear formwhich has the following properties

(I) B is continuous, i.e. 3 M > 0 such that

IB(II, v)1 S Mil 1111 1111 vII II, 'V II, vEl/. (6)

Stochastic models in rough-surface hydrodynamic

(2) B is H-<:oercive in the sense that 3 a > 0 such that

237

B(u, u) ~ allun VuEH. (7)

Then 'V FE H' (the dual space of H), 3 a unique solution u* E H such that

Moreover,

B(u*, v) = F(v) V vEH. (8)

IIu*1I1l S .!. IIFilII' .a

(9)

Returning to problem (I), let V = H M.fl), a Hilbert space. Then L, Li E L( V, V') arecontinuous linear operators from V into its dual space V'. In this case, a bilinear from Bcan be defined as

B(u, v) = [Lou, v]vxv- = L (Lou)vdn

= r h3\lu' '1vdfl.JI/ (10)

It is easy to verify that the conditions in eqns (6) and (7) for B( -, .) are satisfied witha = h~, M = m~ h3• Thus by the Lax-Milgram theorem, if Pj is a solution of LoPj =]j,

a

]j E V' then Pj E Vand IIPjllv s.!. 11]j1lv-. Furthermore LiPj E V'; so the right-hand sidesa

of eqn (4) are in V', and each of the equations has unique solution. Denote

M" = max h,il

110 =-a

( 11 )

and set 11011 ('0 = m~x lo(x, y)1 = 1 as a normalization. By the Cauchy-Schwartz inequality.II

Likewise

(l2a)

IIL2ullv- ~ 3Mhllull v. IIL21IL(v,Y") S 3Mh = 12

IIL3ullv- S lIullv, IIL31IL(v.Y") S I.

(l2b)

(l2c)

We shall use the a priori bounds to establish convergence properties of the expansionsfor p and (p).

4. If I CONVERGENCE FOR P

By virtue of eqns (9) and (12), we can show by mathematical induction (see theAppendix) that

( 13)

where 1= max(ll, 12• h)O. Thus. if E < ~ll' series eqn (3) will converge to P strongly1+ 0

in V = H6(n).

238 S. R. WU and J. T. ODEN

5. H I CONVERGENCE FOR (P)According to the fundamental properties of random variables and random processes

(c.f. [8]), expectations (u) satisfy

~ (u) = (au) .ax axHere

ex.

(P) = L En(Pn)

IJ=O

Then

'" '" ox,II(L Enpn)IIJlI S L Enll(Pn)II/l1 S L En(IIPnll~,)1/2IJi:3 n>3 hi:3

( 14)

(15)

co

~ L En«(lo/(l + 10l)n-3)2(IIPollv+ IIP.llv+ IIP21Iv)2)1/2. (16)n:.3

Recalling eqns (II) and (12), (0 and I are constants independent of the random process.Thus,

co

II(L Enpn)llu' S L En(1+ 1ol)n-3/0/(IIPolll' + liP, IIv + IIP2I1v)2)1/2. (17)n:.3 n~3

Hence eqn (14) will converge to (P) strongly in H'(fl) [as does eqn (3) for P], when1

E < 1 + 10/'

6. CASE 2: C2 CONVERGENCE WHEN 0 E CI(O)

If u is smooth enough, we have now Lju E CO(fl), but no more. According to theregularity theory for elliptic equations (d. [9]), ifjE CO(Q), there exists a unique solutionu E C2(fl) such that Lou = land Ilullc2(ll) S C(11/11co + 114>lI(2). 4> being the boundaryvalue, which here is zero. By the definition of Courant [9], 114>llc2 = 0, we denote 10 = C,

lIullc2 S 1011/11('0.Also Lju E CO(fl).

IIL,ull(~, = max 1-3h20'12u - 3'1(h20)· 'lui S 1,(It, 0)lIullc2il

IIL2ulico = max 1-3h02'12u - 3'1(h02)''1ul S Mh, 0)lIullc2il

(18)

( 19)

The system in eqn (19) is similar to eqn (12), therefore eqn (13) is still valid, if wedenote V = C2(O). Therefore, we have C2 convergence of the expansion for P when

IE<I+lol°

Stochastic models in rough-surface hydrodynamic 239

Since, in this case, the ItS involve 0, h and their derivatives, relations similar to eqn(16) now become very complicated. To eliminate the dependence of the random process,we shall make an additional assumption, which seems to be generally reasonable. InReynolds roughness theory, it is customary to assume that the ratio of wavelength to the

film thickness ~ > !or 1, i.e. ~ < 3 or 5, or some appropriate constant. If we continue

to normalize 0, lolco = max 101 = I, then we may add the assumption:il

(Ii): IVul s M

uniformly for any normalized II in the family of roughnesses considered here.This assumption implies that the roughness (e.g. asperities) rather smooth, and this

allows us to write, by virtue of eqn (19), IILil1 s Ii s I, independent of the randomprocess. Note that

m~x l(u)1 = m.!lX IIUdfl s I m.!lX luldf= (m.!lX lui)a II a a

where f is the stochastic distribution function for t~e random process. Thus

\I(u)11cz= m~x [1(u)1 + L (la(u)1 + la2(u)I)]a

s m~x l(u)1 + L (m~x l(au)1 + m~x l(a2u)1>II II II

s (m~x lui) + L «m,ax laul) + (m~x \a2111» = (11I1I1e2). (20)II II II

Then we obtain

II( L f"Pn)llc2 ~ L Enll(Pn)llc2 S L E\IIPnll)c2

oJ 03 03

~ L En(1 + lol)n-3/01(\IPo\lc2 + IIPtllc2 + \IP2Ib). (21)n.. 3

ITherefore, when E < I + 101' we can also have the C2 convergence of eqn (14) for (P).

7. CONCLUSION

Here we have shown that the series eqns (13) and (14) converge strongly in the H·norm when 0 E CO(O), and in the C2 norm when 0 E C'(O), provided assumption (H)holds. This allows us to attribute a special mathematical significance to these expansions,i.e. these are no longer formal expansions, but have meaning as a sort of geometric seriesin H '(fl). In particular, the residual of an n term-approximation of p will be of orderO[En(l + 10/)"], and this observation provides richer information on the order 0(;)accuracy of the approximation of Po for (P) than was possible in previous work.

Acknowledgement-Research sponsored by the Air Force Office of Scientific Research (AFSC), under ContractF49620-84-C-0024. The United States Government is authorized to reproduce and distribute reprints forgovernmental purposes notwithstanding any copyright notation hereon.

REFERENCES[I) S. T. TZENG and E. SAIBEL, On the effects of surface roughness in the hydrodynamic lubrication theory

of a shon journal bearing. Wear 10, 179-184 (l967a).(2) S. T. TZENG and E. SAIBEL. Surface roughness effects on slider bearing lubrication. ASLE Trans. 10,

334-338 (1967b).[31 H. CHRISTENSEN, Stochastic models for hydrodynamic lubrication of rough surfaces. Proc. Insc. Mech.

Engrs. 184, 1013-1022 (1969-70).

240 S. R. WU and J. T. ODEN

(4] S. M. RHODE and D. WHICKER, Some mathematical aspects of the hydrodynamic lubrication of roughsurfaces, Proc. 4th Leeds-Lyon S}'mp. on Tribology (Edited by D. Dawson et al.). Mechanical EngineeringPublications (1978).

(5) N. PHAN-THIEN, On the mean Reynolds equation in the presence of surface roughness: squeeze filmbearing. 1. Appl. Mech. 48, 717-720 (1981).

(6) N. PHAN-THIEN, On the mean Reynolds equation in the presence of homogeneous random surfaceroughness. J. Appl. Meeh. 49, 476-480 (1982).

(7) I. BABUSKA and A. K. AZIZ. Survey lectures on the mathematical foundations of the finite elementmethod, in The Ma/hemalical Foundations of the Finite Element Melhod with Applications to PartialDifferential Equalions (Edited by A. K. Aziz), pp. 5-359. Academic Press, New York (1972).

[8] A. PAPOULIS, Probability. Random Variables and SlOchastic Processes. McGraw-Hili, New York (1965).(9) R. COURANT, Methods of Malhematical Physics. Vol. 2. Interscience (1962).

[101 J. T. ODEN, Applied Functional Analysis. Prentice-Hall. New Jersey (1979).

(Received 25 June 1984)

APPENDIXProof of eqn (13)

To simplify notations we denote q, = IIPill,. And by the regularity eqn (19) or eqn (18) we denote lIulis 10111 from the equation Lou = /. Thus from eqn (4). LoP, = -L,Po - L2P. - L, P2, and we have

(n = 3 case).

Similarly,

(n = 4 case).

UPsu s Io/(IP,. + IP.U + q2)

s 100[q, + Io/(C/o + q, + q,) + 10/( I + lolXC/o + q. + q,»)

s 101[( I + 100Xqo + q, + q,) + 10/( 1 + 100Xqo + q, + q2))

= 10I( I + 100)1(qo + q. + ql) (n = 5 ease).

We have verified the starting cases n - 3-5. Now suppose that eqn (13) is valid for n ~ 5, then

Ip ... Ms loi(MP.1l + "P._II + IP.-,D) s (/01)'(1 + 100r-S«1 + 101)' + (I + 101) + I](qo + q, + q2)

(I + 1tI» - I= (101)'( I + Iolr-s l'l (qo+ q. + q2) s JoI( I + Iolr-S<3(qo+ q, + q2)'

(1+0/)-1

That means that eqn (\ 3) is valid for N + I. Thus eqn (13) is proved by mathematical induction.