/'" J. £"R"g .'1<'1 Vol. 23, ~o, 2, pp. 207-215, 1985Printed in tb< U.S.A.

0020-7225/85 noo •. ooe 1985 Perpmon rre.s ltd.

EXISTENCE OF SOLUTIONS TO THE REYNOLDS'EQUATION OF ELASTOHYDRODYNAMIC

LUBRICA TION

J. T. ODEN and S. R. WUTICOM. The University of Texas at Austin. Austin. TX 78712. U.S,A.

Abstract-In this article. we prove theorems establishing the existenc:e of solutions to a highlynonlinear variational inequality arising in the study of the flow of an incompressible NeMonianlubricant between elastically-deforming be<lrings. The problem is formally equivalent to Reynolds'equation in elaslOhydrodynamic lubrication theory. The key to the analyses of this class ofproblems is thai they are characterized by pseudomonotone opc:rators. which means that establishedresults on pseudomonotone variational inequalities are applicable.

I. INTRODUCTION

WHEN A viscous lubricant is forced to flow between elastic bearings. substantial clasticdeformation of the bearings can take place which. in turn. aftccts the flow field of thelubricant. A classical model of this phenomena is provided by Reynolds' lubricationtheory in which the film thickness H of the lubricant is sought. In the so-calledelastohydrodynamic lubrication (EJ-IL) problem. the film thickness depends upon theelastic deformation of the bearing surface. In the study of such problems. it is customaryto relate the film thickness H to the hydrostatic fluid pressure p using Hertzian contacttheory.

The present investigation focuses on the question of the existence of solutions to thenonlinear boundary-value problem arising from this elastohydrodynamic-Hertzian theory.We begin by introducing a special formulation of this problem. which is characterized bya nonlinear variational inequality. In this respect. the problem is similar to that of Hertz-contact in the sense that each can be formulated as a variational inequality with aunilateral constraint. Here. however. the similarity ends. for the present formulationinvolves a highly nonlinear pseudomonotone operator. On the other hund. the generaltheory of variational inequalities discussed in this journal by aden and Kikuchi [I]covers cases of non-monotone operators and is shown here to be applicable to thisparticular class of problems.

The results recorded here should prove to be useful in the numerical analysis of suchlubrication problems and. particularly. in the derivation of error estimates and criterialor convergence and numerical stability,

2, THE CLASSICAL PROBLEM

We consider a spherical or cylindrical clastic bearing rolling at an angular velocity w

(about a y-axis through its centroid) with a velocity component U in the x-ctirection.parallel to a rigid surface S. The bearing is separated from S by a film of lubricant whichis assumed to be a viscous. isotropic. incompressible fluid. The (initial) geometric gapbetween the bearing and S is given as a function 11 ~ ho > O. which is usually assumedto be quadratic in x (in the case of a ball or roller bearing) or linear in x (in the case ofa thrust or slider bearing). The film thickness H is unknown and depends upon hi. theelastic deformation of the bearing surface. Some of these quantities arc illustrated inFig. I.

The Reynolds' equation for steady-state motion of an isothermal incompressible.Newtonian lubricant is of the form

207

208 J, T. ODEN and S, R. WU

z

u

sFig. I, Bearing geomelry.

x

"p>O

p = 0 in no.Plan = 0

H = h + h,

(1)

whcrc p is the pressure. µ the viscosity. n is a regular domain supposed to be largeenough to contain the contact domain n,. which is also assumed to be regular.

It is customary in the analysis of problems of this type to assume that the lubricationpressure imparts to the bearing thc same deformation as that under static loading. Thcnthe elastic component of the film thickness is given as the usual Hel1z-contact solutionlinearly in terms of p

h,(x, y) = r K(x. y, ~, TJ)p(~. 1) d~ d1)In (2)

{

Ixo - ~I -C1 In I I (xo E nl)x-~

K(x, y. ~. TJ) = I

C, ,I 2 '- v«x -~) + (y - 1)-)

(for line contact)

(for point contact).

Thus. by introducing Eq. (2) into the expression for Fl. we have

H = H(P). H(P)(x. y) = h(x. y) + r k(x. y. ~. 1))p(~. 1) d~ d1).JIl (3)

Denoting by A the nonlinear operator

and by fthe given function

f= -6µ iJ(Uh)ax

the classical EHL problem for the pressure assumes the form:

(4)

(5)

Existence of solutions 10 the Reynolds' equation of elastohydrodynamic lubrication 209

Find a pressure field p such that

{

Ap = f,

p=o

pl"u = o.

p>o

in noin nl

(6)

3. VARIATIONAL INEQUALITY

Let V denote the Sobolev space

{ av av IV=H6(n)= v.-,-EL2(n) v=oax ay

and V* its topological dual

on an (in the sense oftraCCS)} (7)

(8)

We denote by ( ' .. ) duality pairing on V* X V. Further. let K denotc the closed convexsubset of V defined by

K = {v E vlv ~ 0 (a.e, in n)}. (9)

Then the opcrator A of (4) can be considered as a map from V into V'.As a variational formulation of problem (6). wc consider the following variational

inequality:Find p E K such that

(Ap - f, q - p) ~ 0 Vq E K. ( 10)

The relationship of Eq. (10) to Eq. (6) is made clear in the following proposition.

Proposition ISuppose that there exists a solution p to problem (10). Then p is a distributional

solution to Eq. (6): i.e. p ~ 0 in HI(Q). Ap - f= 0 in a distributional sense in nl. P = 0a.e. in no and p > 0 in n,.

Proof Denote n} = {(x, y) E nlp(x. y) > O}. no = {(x. y) E nlp(x. y) = O}. Let I/;E CO'(nd be such that I/; > 0 in fll and I/; = 0 in ~. Denote q = p + fl/;, Then Eq. (10)implies that f(Ap - f, 1/;) ~ O. Since f is arbitrary. this implies that Ap - f= 0 in no (ina distributional sense). The constraint on p follows from the fact Ihat p E K. 0

Remark: a necessary condition 011 the data.f Returning to Eq. (10), if p = 0 in no. Ap= fin Ql. we then have for smooth q E K. (Ap - f, q - p) = - fl10fq dn ~ O. Since q~ O. it is necessary that

f~O in flo. (II)

Returning to Fig. I, we observe that we can take U = WI' cos 0, h = ho + 1'( I - cosX 0), cos 0 = V(r2 - x2)/r.! = -6µ(aUhJa;x) = -6µ(wx/Y(r2 - x2»«2 cos 0 - 1)1' - ho).Since in most lubrication problems. 0 -- O. ho ~ r. we will havef~ 0 when x ~ 0 andf~ 0 when x s; O. That means no should be in the half-plane x ~ 0, Thus. without addingthis condition to the data in problem (6). it will, in general. be impossible to establishthe usual equivalence between problem (6) and the variational inequality. It is interestingto note that numerical solutions 10 problem (6) almost always exhibit 'contact' regionssuch that Eq. (11) holds (see. e.g. [2]).

4, EXISTENCE THEOREM

An operator A mapping a Banach space V into its dual V* and a sequence {um} E Vis said to satisfy conditioll P if and only if

210 J. T. ODEN and S, R. WlJ

11m ~ II (weakly in V)

lim sup (A(lIm). lim - II) ~ O.01-00

(12)

An operator A: V - V* is pselldomo/lorone if and only if. whenever A, {11m} satisfycondition P. we have

lim inf (A(lIm). 11m- v) ~ (A(II). II - v)nJ-OO

"Iv E V. ( 13)

The operator A: V - V* is hemiconrinllolls if and only if t{;(t) = (A(II + TV). 1\') is acontinuous function of t E [0. I] for all lI. v, IV E V.

The following existencc theorem can be established for variational inequalities involvingpseudomonotone operators:

Proposition 2Let K be a non-empty. closed, convex subset of a reflexive Banach space Vand let A:

K - V* be a pseudomonotone. bounded, and coercive operator from K into the dualV* of V, in the sense that there exists Va E K such that

(Av, v - vo) = +00.lim IIL'IJ11"11-'"

Let fbe given data in V*. Then there exists at least one II E K such that

..

(All - f, V - II) ~ 0 "Iv E K. ( 14)

ProQ{ The proof of this theorcm can be found in Ref. [I]. For a complete discussionof pseudomonotolle variational inequalities. their history and pertinent references, see thebook of aden [3], 0

Remark. To establish that a given non-monotone operator is pseudomonotolle. it issufficient to show that it is a completely continuous perturbation of a monotone operatoror. in particular. if it satisfies a generalized Giirding inequality. See aden [3. 4]. 0

Our major result is embodied in the following theorcm:

Theorem I

Let f satisfy Eq. (II), Then there exists at least one solution p E K to the variationalinequality (10).

Our proof of this result shall be broken into several lemmas which. collectively.establish that the conditions or Proposition 2 are satisfied.

Lemma IFor hI defined in Eqs. (3) and (4). 0 < (X < I. q = (2 - a)/( I - a) > 2. 3 C> 0 such

that

ma~ Ih)(P)1 ~ CIIPIIL.(x.y)Ell

VpE K. ( 15)

PrOl~r Recall that hI = III K(x . .1', ~. 7j)P(~. 7j) d~ d1). For case (I)

V(x, .1') E Q. p> O.

For case (2)

0<(\'<1. V(x. y) E Q.

Existence of solutions to the Reynolds' equation of c1aslohydrodynamic lubrication 211

By Holder's inequality. we always have

Ill1d5 IIKI/L2-a(tI(E.~.))(X. y. )I/PI/u1 2-0'

q= =->2.1 1- a

1--2-0'

It is evident that in case (I) /lKl/u-u is continuous on Q. In case (2)

In IK(x, y. ~, 1])12-"d~ d1]

(16)

{l2.: lR(81 r dr C? l2 ..

dO C2 --:;=- = --= R"(x, y. 0) dO.o 0 r-" a 0

.. R(e) r dr C 'IIr dO r C2 2=- = --1 r R"(x, y. 0) dO.Jo Jo r a a Jo

if

if

(x. y) E int !l

(x. y) E an(17)

R(x. y. 0) being the distance from (x. y) to au. It can be shown that IIK/lu-u = Uti IK(x.y. ~. l1)12-ad~ d1]]I/(2-a) is continuous on Q by examining Ra.

If we denote M 1\ = max IIK 11/.2-0. we haveii

max 1111(1')1 ~ MKIPI//. •.!!

( 18)

Lemma 2The operator A of Eq. (4) is bounded as a map from V into V*.Proqf By defining (AI', q)v.",* = III (-v' (ff3(p)v P) + 6µ(d( Uh,(P»/d.\)]q dr? = III

X (Ff3(p)v p. vq - 6µuh,(p)(aq/ax)] dr? 'ltp. q E K. Denote

Then by virtue of Eq. (14).

Mil = max lUI,ii

M" = max Ihl.!I

(19)

mfix IH(p)1 = max 111+ 11,(1')1 5 M" + l\hI/PIII..II ii

(Ap. q) 5 (Mh + MI\\IP\lL.)3 r IvP' vql d!l + 6µMuAh\lP\lu r Idql dQJll Jo a\5 (M" + MKIIPllu)31Iplllldlqllll' + 6µMui\'h\lPIIL.Collq/l1l1.

Thus we obtain

By the Sobolev embedding theorem (cf. Ref. [5]. for 11 = 2. 111 = L p = 2 case. mp= 11.We have HI(n)L. U(n) compactly. Then 3 Co> 0 such that

(20)

Therefore, IIAp/I V' 5 (M" + Col\hllPl/lld31lPIIIII + 6µCuCoMui\hllpIiN"That means that A maps the bounded set in K into bounded set in V*. So A IS

bounded. 0

212 J, T. ODEN and S. R. WU

Lemma 3The operator A of Theorem I is coercive (with Vo = 0 E K)PrOl~r.

(All, v) = L [I/3(v)lV'vI2 - 6µh,(v) ::] dQ

= In [h(V)H2(11)IV'VI2 + hl(v)(H2(v)lV'vI2 - 6µ ::)] dQ

~ MC'llvIIJf' + In hl(v)[hfi(v)IV'lf - 6µ1V'111JdQ.

Here we have used Poincare's inequality

(21 )

3 C> 0, (22)

then.I

C' = I + C2'

Let

Q2 = n - QI' The integrand in Eq. (21) is positive in Q2 and negative in nl. Then

Hence

(Av, v) --. 00

IIvii as Ilvll --. 00. D

Lemma 4For any Ii, v E V such that IIIi III/I, IIV III/I =:;; 1/ there exists a constant C > O. such that

Proof We immediately have.

(Ali - Av, II - v)

r [ a(1i - V)]= J{! f/3(1i)\11i' \1(/1 - v) - H3(v)\1v, \1(/1 - v) - 6µUh.(u - v) ax dQ

a(1i - V)] L- 6µUhl(u - v) dQ ~ Clh~llu - vIIi/I + (H(u) - H(v»(1l2(u) + H(u)H(v)ax {!

Existence of solutions to the Reynolds' equation of e1astohydrodynamic lubrication 213

+ H2(v»'Vv· 'V(II - V) - 6µA.f.~hllll - vllu L I'V(II - v)1 dQ

~ Clh~111I- v1l11' - Mdlll - vIILq2[(Mh + 1\hIlIlIILq)2

+ (Mil + AhllvllJ.q)2] '11vlllllllll - villi' - 6µM,,1\hllli - vllJ.qColili - vlllll. (24)

For 111111111, Ilvlllll ::; 1]. by virtue of Eq. (16) we obtain

Lemma 5A is hemicontinuous. that is 'VII. v, wE V. I > 0

lim (A(II + IV), 11') = (All. 11')./-0'

Proof

L[ aw](A(II + tv). 11') = H3(u + IV)'V(u + tv)· 'VII' - 6µUhl(1I + IV) -: dHo a~

(26)

L [a"']- H3(1I)'VII' 'VII' - 6µUhl(lI) -: dH = (All. w).II a.\

Lemma 6The operator A is pseudomonotone. Now we will prove (by the definition of

pseudomonotonc operators) that if II" - li weakly in K. lim sup (All". II" - II) :::;O.then lim inf (All", II" - v) ~ (Ali. II - v) 'Vv E V. "-00

(I) Since II" - II weakly in the reflexive Banach space V = /lb(n). {u,,} is bounded.Let lIu"lllI' ::; 1]. lIulllll ::; 1]. From Eq. (25). we have

(27)

According to Sobolev embedding theorem, we have HI C. Lq compactly. SO II" ~ ustrongly in U. Thus we obtain from Eq. (27)

o ~ lim (Au". Un - u) ~ lim (Au", II" - II)

~ lim [(Au. u" - u) - Cllu" - ullLqllu" - ulillo] = 0

lim (All", li" - II) = O.II-a:;

(2) 'VII' E V. like Eq. (24)

(28)

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

= J {[III(II" - II) + hl(u - w)][(H(u) + 1I\(u" - 11»2 + (l1(u)

+ h,(u" - u»1f(w) + H2(w)]vw-(v(u" - II) + V(U- w»

(iJ(lIn - u) iJ(u - W»)}

- 6µU(III(II" - u) + 11,(11 - 11'» aX + ax dn

= '\ + h + 13 + 14 + 15

f [ iJ(u - W)]I, = 1I1(lIn - II) (I/2(lIn) + H(II,,)H(w) + H2(II'»vl\'- V(II" - 1\') - 6µU n _ dn

II ax

12 = L [111(11 - W)(1f2(U) + H(u)H(w) + 1f2(II'»VW- V(II - IV)

iJ(1I - W)]- 6µUht(u - 11') ~. dn.

h = f h\(11 - w)(1I2(u) + 1/(1I)I1(w) + /-I2(W»VW- v(u" - u) (inJIl14 = L hl(1I - w)lIl(u" - u)(2H(u) + 11\(11"- u) + l/(II'»VW- V(II" - w) dQ

i ' a(Un - II)'5 = - 6µ.l.h\(11 - w) dU.Il ax

Since by Eq. (18) m<;lxIhl(lI" - 11)1~ J\hllll" - III1Lq - O. II --I O. /4 --I 0, taking intoIl

account of the houndedness of the remaining terms in II and 14'

13 = (-v -(- . '). II" - II) ~ 0

_ ( d(lJh,(1I - w» )15 - 6µ iJ ' u" - u - O.x

Since u" --I u weakly in III.Thus we obtain from the same procedure as from Eq. (24) to Eq. (25)

(3) In view of Eqs. (28) and (29)

lim (Au". II - 11') = lim (Au". II" - w) ~ lim (Aw, II" - w)- CIlII - 1\'11[.qllll - 11'11111 = (All'. II - w) - C1lu - 1\'III.qllll - 11'11/11. (30)

(4) 'VvE V, let I\' = II + 8(v - u).O < 0 < I. for small 8 Ilwll < 71.

Substituting II' in Eq. (30). results in

lim (Au", O(u - v» ~ (A(II + O(v - 11».0(11 - v» - CfPIIII - vll/.qllu - viiI/I.

Dividing both sides by 8 and letting 0 - 0: we obtain the desired result by virtue of Eq.(26)

lim (Aun. u - v) ~ (All. II - v).

Thus the proof is completed.Remark (1Il1iqueness). Using Eq. (20) in Eq. (23). we have (All - Av. II - v) ~ (CII~

- CCo)1I11- vll~,. Thus. if Ch~> Cco. A is stricti}' monotone. Then the solution of Eq.

Existencc of solutions to the Reynolds' equation of elastohydrodynamic lubrication 215

(6) is unique. However. if C'h~ ~ CCo. we can conclude nothing about uniqueness fromour analysis. More specific conditions for uniqueness of solutions or the developmentof criteria for bifurcation corresponding to instabilities of the lubricant film needfurther study.

AckIlOll'led,::emelll- The support of this research by the Air Force Office of Scientilic Research (AFSC), undcrcontract F49620-84-C-0024 is gratefully acknowledged.

REFERENCES[II J. T. ODEN and N, KIKUCHI. Int, J ElIglIg Sci. 18. 1173 (1980).[2} B. J. HAMROCK and D. OOWSON, Ball Bearillg l.ubrication, the ElaslOhydrodYllamics of Elliptical

Contacts. Wiley. New York (1981),[3] J. T. ODEN, Qualitatil'e Methods ill Illol/lillear MedulIlirs. Prentice-Hall. Englewood Cliffs. New Jersey

(1984).[4] J. T. ODEN. J. Math, Allal. alld Applic, 69( I). 51 (1979).[5} R. A. ADAMS, Sobo/(>I'Spares. Academic Press, New York (1975),

(Reeeh'ed 22 Ma." 1C)84)