Homogenization Theory with Applications in Tribology

LICENTIATE T H E S I S

Luleå University of TechnologyDepartment of Mathematics

2008:28|: 02-757|: -c -- 08 ⁄28 --

2008:28

Homogenization Theory with Applications in Tribology

Universitetstryckeriet, Luleå

John Fabricius

John Fabricius H

omogenization T

heory with A

pplications in Tribology

2008:28

Homogenization Theory with Applications in

Tribology

John Fabricius

Department of MathematicsLulea University of Technology

SE-971 87 Lulea, Sweden

Key words and phrases. Mathematics, Homogenization theory, Partial differentialequations, Calculus of variations, Multiscale convergence, Two-scale convergence,Asymptotic expansion method, Tribology, Hydrodynamic lubrication, Reynolds

equation, Surfaces roughness

To the memory of Carl-Ake

Abstract

This thesis is devoted to the study of some homogenization problems withapplications in tribology. Homogenization is a mathematical theory for studyingdifferential equations with rapidly oscillating coefficents. Many important prob-lems in physics with one or several microscopic length scales give rise to this kindof equations. Hence there is a need for methods that enable an efficient treatmentof such problems. To this end several homogenization techniques exist, rangingfrom the fairly abstract ones to those that are more oriented towards applications.This thesis is concerned with two such methods, namely the “asymptotic expansionmethod”, also known as the “method of multiple scales”, and multiscale conver-gence. The former method, sometimes referred to as the “engineering approach tohomogenization” has, due to its versatility and intutive appeal, gained wide accep-tance and popularity in the applied fields. However, it is not rigorous by mathe-matical standards. Multiscale convergence, introduced by Nguetseng in 1989, is anotion of weak convergence in Lp spaces that is designed to take oscillations intoaccount. Although not the most general method around, multiscale convergencehas become widely used by homogenizers because of its simplicity. In spite of itssuccess, the multiscale theory is not yet sufficiently developed to be used in connec-tion with certain nonlinear problems with several microscopic scales. In Paper Awe extend some previously obtained results in multiscale convergence that enableus to homogenize a nonlinear problem with three scales. In Appendix to Paper Awe present in more detail some results that were used in the proofs of some of themain theorems in Paper A.

Tribology is the science of bodies in relative motion interacting through a me-chanical contact. An important aspect of tribology is to explain the principles offriction, lubrication and wear. Tribological phenomena are encountered everywherein nature and technology and have a huge economical impact on society. An im-portant example is that of two sliding solid surfaces interacting through a thin filmof viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressuregenerated within the lubricant, through the viscosity of the fluid, is able to sus-tain an externally applied load. Many common bearings, e.g. journal bearings orslider bearings, operate according to this principle. As a branch of fluid dynamics,the mathematical foundations of lubrication theory are given by the Navier–Stokesequations, describing the motion of a viscous fluid. Because of the thin film as-sumption several simplifications are possible, leading to various reduced equationsnamed after Osborne Reynolds, the founding father of lubrication theory.

The Reynolds equation is used by engineers to compute the pressure distribu-tion in various situations of thin film lubrication. For extremely thin films, it hasbeen observed that the surface microtopography is an important factor in hydro-dynamic performance. Hence it is important to understand the influence of surface

v

vi ABSTRACT

roughness with small characteristic wavelength upon the pressure solution. Sincethe 1980s such problems have been increasingly studied by homogenization theory.The idea is to replace the original equation with a homogenized equation where theroughness effects are “averaged out”. One problem consists of finding an algorithmthat gives the homogenized equation. Another problem, consists of showing, byintroducing the appropriate mathematical defintions, that the homogenized equa-tion really is the correct one. Papers B and C investigate the effects of surfaceroughness by means of multiscale expansion of the pressure in various situations ofhydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis,considers homogenization of the stationary Reynolds equation and roughness withtwo characteristic wavelengths. This leads to a multiscale problem and adds to thecomplexity of the homogenization process. To compare the homogenized solutionwith the solution of the unaveraged Reynolds equation, some numerical examplesare also included. Paper C is devoted to homogenization of a variational principlewhich is a generalization of the unstationary Reynolds equation (both surfaces arerough). The advantage of adopting the calculus of variations viewpoint is that therecently introduced “variational bounds” can be computed. Bounds can be seenas a “cheap” alternative to computing the relatively costly homogenized solution.Several numerical examples are included to illustrate the utility of bounds.

Preface

This thesis is composed of three papers and a complementary appendix. Thesepublications are put to a more general frame in an introduction which also servesas a basic overview of the field.

A. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Reiterated homog-enization of a nonlinear Reynolds-type equation. Research Report, No. 4,ISSN:1400-4003, Department of Mathematics, Lulea University of Technol-ogy, (19 pages), 2008.

Appendix to A. J. Fabricius.

B. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Reiterated homoge-nization applied in hydrodynamic lubrication. To appear in Proc. IMechE,Part J: J. Engineering Tribology, 2008.

C. A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Variational boundsapplied to unstationary hydrodynamic lubrication. Internat. J. Engrg.Sci., 46(9):891–906, 2008.

vii

Acknowledgement

I thank my main supervisors, Prof. Lars-Erik Persson and Prof. Peter Wall,for guiding me in the fascinating field of mathematics, for giving encouragementand support in times of need and for always showing a generous attitude towardstheir students.

I also thank Dr. Andreas Almqvist and my fellow Ph.D. student EmmanuelEssel, with whom collaborating has been “frictionless” and much joy. Dr. Almqvistalso accepted the role as my co-supervisor, contributing with his expertise in tri-bology. For this I am also very grateful.

During my undergraduate studies at Uppsala University and ENS Lyon I hadthe privilege to have several inspiring teachers who influenced both my views onmathematics as well as my taste in this subject. Finally, a special mention goes tomy colleagues at the Department of Mathematics who do an excellent job promotingmathematical activity in the vicinity of the Arctic Circle. You light up the darkpolar night.

ix

Introduction

Tribology is the science of interacting surfaces in relative motion. The wordtribology is derived from the Greek tribos which means ‘rubbing’. Many machinecomponents, e.g. bearings, gears, piston rings, tyres, breaks and magnetic storagedevices, consist of parts that operate by rubbing against each other. Tribologyis an interdisciplinary science dealing with such diverse phenomena as friction,wear, lubrication and contact mechanics. Any mechanical system in which frictionoccurs will experience energy loss in terms of heat dissipation. Wear in machinecomponents may cause mechanical failure and shortens their life span. According tothe Jost Report, issued by the British government in 1966, and subsequent studiesin other countries, the estimated costs due to tribological phenomena are of theorder of one per cent of the gross domestic product. Thus, rational machine designhas huge economical motives.

Lubrication is the action of viscous fluids to diminish friction and wear betweensolid surfaces. It is fundamental to the operation of all engineering machines andmany biological mechanisms, e.g. hip joints. It can be observed that a converg-ing fluid film is able to separate two surfaces in relative motion pressed togetherunder an external load. When a fluid film achieves this state, called the hydrody-namic lubrication regime, solid to solid contact is prevented and the applied loadis supported by a pressure that develops within the film because of the lubricant’sresistance to motion.

In 1886 Osborne Reynolds published a theory of lubrication [66], now considereda milestone in tribology. By applying the principles of fluid dynamics, Reynoldsderived the equation, subsequently reffered to as “the Reynolds equation”, thatgoverns hydrodynamic lubrication of a cylindrical journal revolving in a cylindricalbearing (see Figure 1). The Reynolds equation is a two-dimensional approximationof fluid flow that relies on the assumptions that the surfaces are “nearly parallel”and that the radii of curvature of both bearing and journal are large compared withthe thickness of the film. The Reynolds equation can take a variety of forms, see[36, 69]. If the lower surface, denoted by Ω, is a portion of the plane x3 = 0 and theupper surface is the corresponding portion of the surface x3 = h(x1, x2), where h isa smooth, everywhere positive function, then a simple form of Reynolds equationreads

(1)∂

∂x1

(h3 ∂p

∂x1

)+

∂

∂x2

(h3 ∂p

∂x2

)= 6μv

∂h

∂x1

The unknown p = p(x1, x2) is the pressure distribution and h is commonly referredto as the thickness of the film. One surface is moving at constant speed v in thex1-direction, the other remains fixed. No slip occurs at the boundary of the movingsurface. The lubricant viscosity μ is taken as constant. Equation (1) is assumed

1

2 INTRODUCTION

Figure 1. A schematic description of the journal bearing inTower’s experimental set up (from [66]). The bearing was im-mersed in an oil bath.

Figure 2. Reynolds’ illustration of the principles of hydrody-namic lubrication (from [66]). The upper graph represents thepressure distribution. The lower graphs represent the fluid motion.

to hold in the interior of Ω and the standard boundary condition is to prescribep = 0 on ∂Ω. In the derivation of (1), h is assumed to be small with respect to thedimensions of Ω. Reynolds’ work provided a theoretical basis for the understandingand design of bearings and proved to be in close agreement with a series of carefulexperiments that had previously been conducted by Tower. Although primarilyknown as a prominent figure in fluid dynamics, Reynolds also contributed to otherareas of tribology through his fundamental work on rolling friction.

For a long time, a limitation on the applicability of Reynolds’ theory was thedifficulty of obtaining two-dimensional analytical solutions of (1). Closed-form solu-tions were known only for infinitely long bearings (Sommerfeld 1904) and infinitelyshort bearings (Michell 1905 and Ocvirk 1952). The advent of fast computers,however, has made the Reynolds equation an indispensable tool in bearing design.Compared to more general physical models such as the Navier–Stokes equations,the Reynolds equation is linear, lower-dimensional and gives the asymptotically cor-rect solution as the film thickness h tends to zero. Reynolds’ equation has provento be a reasonable approximation for many problems in hydrodynamic lubrication.The transition between the three-dimensional models for viscous flow and lower-dimensional approximations is a subject that has attracted quite a lot of attention.This should not be surprising as flows in thin domains are encountered in varioussituations, not only in fluid film bearings but also in e.g. gas pipelines, capillaries,

INTRODUCTION 3

oceans and the atmosphere. As shown by many rigorous studies, in the mathe-matical litterature [11, 15, 31, 32, 35, 49, 54, 55] and in the engineering litterature[10, 24, 36, 70, 72], the scaled Reynolds equation gives an O(h) approximation,i.e. a zeroth-order approximation, of the pressure distribution. For lubricationwith high Reynolds number, e.g. lubrication of a rapidly rotating shaft, Reynolds’approximation becomes rather crude. The need for higher-order approximationsof pressure and velocity fields in hydrodynamic lubrication has been confirmed bynumerous theoretical and experimental studies, see the references cited in [50, 56].Recent investigations [50, 53, 56] in this direction have lead to modified nonlinearReynolds-type equations, containing terms that arise from inertial and curvatureeffects. Hydrodynamic lubrication with non-Newtonian fluids is also an active re-search field. A nonlinear Reynolds-type equation accounting for non-Newtonianeffects has been proposed in [41].

Even when the error in the Reynolds approximation is tolerably small, at leasttwo other effects have been recognized that could render the Reynolds equationinvalid. First, there is the effect of molecular slip at the boundary, causing themacroscopic velocity of the fluid near the boundary to deviate from that of theadjacent surface. Such effects become apparent in magnetic storage devices con-sisting of a mechanical head flying over a rotating disk, the lubricant being air.Modifications to Reynolds equation to compensate for this effect have been dis-cussed by Burgdorfer. Second, there is the effect of surface roughness. Technicalsurfaces are never perfectly smooth because of imperfections in the manufacturingprocess. Almost smooth surfaces can only be manufactured at extremely high costs.Roughness usually increases wear and is therefore undesirable in many situations intribology. In tilted-slider bearings and journal-type bearings however, it has beenobserved that the performance can improve hydrodynamically with the addition ofroughness. Hence, deliberately machined roughness (or texture) can also be con-sidered as a design parameter. In the realm of fluid mechanics roughness is usuallyneglected for laminar flow, but when the lubricant film is sufficently thin even smallroughness becomes significant. According to Elrod [37], the first theoretical studiesof surface roughness appeared in the 1950s. For a review of the state of the artthe reader may consult Elrod’s paper [38], chapter 7 of the monograph [74] and thedoctoral thesis [4]. Surface roughness enters the Reynolds equation through thefunction describing the film thickness. The usual ansatz in statistical treatments isthat h = h0+hR where h0 is a function representing the “global film thickness” andhR is a stochastic variable representing the roughness (in deterministic treatmentshR is related to a specific surface description). Since the roughness is random, thesolution of the Reynolds equation must be averaged at some point in the calcula-tions. For this various techniques have been suggested by many authors, e.g. Tzengand Saibel, Christensen, Christensen and Tønder, Elrod, Chow and Saibel, Patirand Cheng and Phan-Thien [71, 26, 28, 37, 39, 29, 62, 65].

There has been some controversy as to applying the Reynolds approximationin lubrication with rough surfaces actually leads to realistic results. Elrod [37]and Sun and Chen [67] think that the Reynolds equation is inadequate when theroughness wavelength ε becomes smaller than or comparable in magnitude with thefilm thickness h, arguing that the basic assumptions used to derive the Reynoldsequation no longer hold. In this case the Stokes equations can be used instead.Elrod has proposed to classify roughness into two categories: “Reynolds roughness”

4 INTRODUCTION

Figure 3. The sliding of a rough surface against a smooth plane,the arrow indicating the direction of motion.

and “Stokes roughness”. Roughness is said to belong to the former category ifh ε. Early studies [67, 64] showed that the Reynolds equation and Stokesequations may lead to conflicting results, but an explanation to this is difficultbecause of the various assumptions introduced. A rigorous attempt to clarify theconcepts of Reynolds and Stokes roughness is that by Bayada and Chambat [14](see also [13, 51]). Using homogenization theory they study the Stokes equations ina thin domain bounded by a periodically rough surface and a smooth plane whenε and h tend to zero. Depending on the value of the parameter λ = h/ε differentequations are obtained in the limit. Three situations apply:

(1) λ = constant. A two-dimensional equation is obtained with coefficientsthat depend on the surface microtopography and λ.

(2) λ → 0 (Reynolds roughness). Leads to the homogenized Reynolds equa-tion, i.e. first h → 0, then ε → 0. In agreement with previous studies.

(3) λ → ∞ (Stokes roughness). A simple Reynolds equation is obtained withan effective film thickness. The same result is obtained if first ε → 0, thenh → 0.

A remarkable conclusion of this study is that there is really no need to considerthe Stokes roughness. This message does not seem to have assimilated into theengineering community, where more and more attention is given to the Stokesroughness, see the recent studies [59, 46]. We can only speculate why this is so,but perhaps the hypotheses about the roughness in [14] is considered to be tooweak. This also suggests that more rigorous studies of thin film lubrication withtwo rough (curved) surfaces are needed.

In the mathematical community, homogenization has become the dominantapproach to studying the influence of surface roughness in lubrication. Homog-enization theory is a set of mathematical techniques that are aimed at studyingdifferential operators with rapidly oscillating coefficents, equations in perforateddomains or equations subject to rapidly alternating boundary conditions. The firstproof of a homogenization theorem was obtained by De Giorgi and Spagnolo around

INTRODUCTION 5

1970. There exists a vast litterature on homogenization theory, see e.g. the books[3, 19, 20, 25, 33, 34, 43, 47, 52, 60, 61, 63, 75].

Let us qualitatively describe homogenization in the context of equation (1) andsurface roughness. The roughness enters the Reynolds equation through the filmthickness h. It is assumed that the roughness is periodic with period ε. To this endlet h1 be a function that is 1-periodic in both arguments and has mean value zeroon the corresponding cell of periodicity. The film thickness entering the Reynoldsequation is given by

hε(x) = h0(x) + h1

(x

ε

)resulting in the equation

(2)∂

∂x1

(h3

ε

∂pε

∂x1

)+

∂

∂x2

(h3

ε

∂pε

∂x2

)= 6μv

∂hε

∂x1.

Homogenization of (2) has been studied by Wall [73] who showed that the sequenceof solutions pε converges to a limit p0 which is the solution of an “averaged” orhomogenized equation,

(3)2∑

i,j=1

∂

∂xi

(aij

∂p0

∂xj

)= 6μv

∂h0

∂x1−

2∑i=1

∂bi

∂xi.

The averaged or homogenized coefficients aij and bi (1 ≤ i, j ≤ 2) are found byfirst solving v0, v1 and v2 from the three periodic problems

∂

∂y1

(h3 ∂v0

∂y1

)+

∂

∂y2

(h3 ∂v0

∂y2

)= 6μv

∂h

∂y1(4a)

∂

∂y1

(h3 ∂vi

∂y1

)+

∂

∂y2

(h3 ∂vi

∂y2

)= −∂h3

∂yi(i = 1, 2),(4b)

where h(x, y) = h0(x) + h1(y). Then use the averaging formulae

(5)(

a11 a12 b1

a21 a22 b2

)=

∫Y

h3

(1 + ∂v1

∂y1

∂v2∂y1

∂v0∂y1

∂v1∂y2

1 + ∂v2∂y2

∂v0∂y2

)dy,

Y denoting the cell of periodicity, to compute the coefficients. This gives thefollowing homogenization algorithm:

(1) Solve the local problems (4).(2) Compute the homogenized coefficients (5).(3) Solve the homogenized Reynolds equation (3).

Some mathematically delicate questions that arise are in what sense pε convergesto p0 as ε → 0 and whether the coefficients (5) are sufficently “nice” for (3) to bewell posed. Heuristically, one can find the homogenized equation (3) and the localproblems (4) by making the ansatz

pε(x) = p0

(x,

x

ε

)+ εp1

(x,

x

ε

)+ ε2p2

(x,

x

ε

)+ · · ·

and plug this into (2). Equating terms of equal powers of ε and solving the ob-tained system of equations eventually yields the homogenization result. This is theasymptotic expansion method.

The homogenized Reynolds equation (3) contains no oscillating coefficents.Nevertheless it retains local information of the film thickness, thereby capturing

6 INTRODUCTION

the effects of roughness. It is in general different from the Reynolds equation cor-responding to mean film thickness, i.e.

∂

∂x1

(h3

0

∂p

∂x1

)+

∂

∂x2

(h3

0

∂p

∂x2

)= 6μv

∂h0

∂x1.

For general two-dimensional roughness the coefficients of the homogenized equationcan be rather cumbersome to compute. Only in special cases, e.g. the case of one-dimensional (striated) roughness, closed-form solutions of the local problems canbe found.

The first publication that uses the word ‘homogenization’ in the context ofhydrodynamic lubrication is probably [65] which describes a homogenization pro-cedure for Reynolds equation. However, already in 1973, Elrod [37] had proposed avery similar method which he called the “two-variable expansion procedure” or the“method of multiple scales”. Elrod’s work is truly original since it predates homog-enization by the “asymptotic expansion method” of Bakhvalov and Lions. SinceElrod’s pioneering work, ideas from homogenization theory have been frequentlyapplied in tribology. One of the first rigorous studies in the field is that of Bayadaand Chambat [14]. Subsequent studies pertain mostly to Reynolds roughness invarious lubrication regimes, see e.g. the works [44, 16, 17, 23, 30, 18, 73, 6, 48].For homogenization to be useful to engineers in e.g. bearing design, efficient nu-merical treatment of the homogenized equations is needed. Such aspects have beendiscussed in [21, 22]. Comparisons between homogenization and traditional aver-aging of Reynolds equation have been undertaken in [45, 40]. We summarize theirfindings:

• A direct computational approach is limited by the resolution of the nu-merical method employed. A more cost-effective way is to compute thesolution of the homogenized equation even though some local problemsmust be solved first. Actually the finer the roughness, the better theagreement between the homogenized and the deterministic solution.

• Homogenization works regardless of the type of roughness. It is particu-larly well suited for anisotropic roughness, retaining information regard-ing the amplitude and the direction of the roughness. Rival stochasticapproaches prove defective in this case.

• Unlike some averaging methods that may lead to ambiguous results, thehomogenized equation is uniquely determined.

• Being a mathematical theory, homogenization is a completely rigorousapproach, whereas many other methods are based on heuristics.

The papers comprising this thesis [7, 8, 9] are all examples of homogenizationapplied in hydrodynamic lubrication. Paper A [7] focuses on theoretical aspects,whereas Papers B and C [8, 9] are more oriented towards specific applications. Themain result of [7] is a reiterated homogenization result for a nonlinear Reynolds-type equation. To prove this result we first develop some aspects of the theory ofmultiscale convergence introduced by Nguetseng, Allaire, Allaire and Briane andNguetseng et al. [57, 1, 2, 58]. Special attention is given to the linear case. Reit-erated homogenization makes it possible to analyze surface roughness with severalcharacteristic wavelengths. How this is done in practice is explained in Paper B[8], where it is also shown how asymptotic expansion can be used to find the ho-mogenized equation. To compare the homogenized solution with the solution of

INTRODUCTION 7

(a) Deterministic pressure distribution (b) Homogenized pressure distribution

Figure 4. A computer visualization of homogenization ofthe pressure distribution in a rough thin film (courtesy ofAlmqvist et al. [5]).

the deterministic (unaveraged) Reynolds equation, some numerical examples arealso included. Paper C [9] is devoted to homogenization of a variational principlewhich is a generalization of the unstationary Reynolds equation (both surfaces arerough). The advantage of adopting the calculus of variations viewpoint is that therecently introduced “variational bounds”, see [6], can be computed. Bounds canbe seen as a “cheap” alternative to computing the relatively costly homogenizedsolution and give a mathematical explanation to some heuristically proposed aver-aging techniques. For simple type of roughness the bounds even coincide with thehomogenized solution. Several numerical examples are included to illustrate theutility of bounds.

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Paper A

REITERATED HOMOGENIZATION OF A NONLINEARREYNOLDS-TYPE EQUATION

ANDREAS ALMQVIST, EMMANUEL KWAME ESSEL, JOHN FABRICIUS, AND PETERWALL

Abstract. We prove a reiterated homogenization result for monotone opera-tors by means of multiscale convergence. The reiterated homogenization prob-

lem consists of studying the asymptotic behavior as ε→ 0 of the solutions uε

of the nonlinear equation

div aε(x,∇uε) = div bε,

where both aε and bε oscillate rapidly on two microscopic scales and aε

satisfies certain continuity, monotonicity and boundedness conditions. Thiskind of problem has applications in hydrodynamic thin film lubrication where

the bounding surfaces have two types of roughness corresponding to differentlength scales. To prove the homogenization result we extend the multiscale

convergence method introduced by Allaire and Briane [2] to W 1,p0 (Ω), where

1 < p <∞. The original formulation of multiscale convergence pertained onlyto the case p = 2.

1. Introduction

The main contributions of this paper is that some of the previous homogenizationresults in connection with hydrodynamic lubrication are extended to include twomicroscopic scales and non-Newtonian fluids. More precisely, we study the limitingbehavior as ε→ 0 of the solutions uε of

(1)div aε(x,∇uε) = div bε in Ω

uε = 0 on ∂Ω,

where Ω is an open bounded subset of RN and aε and bε oscillate rapdily onthe microscopic scales ε (meso scale) and ε2 (micro scale). The idea of reiteratedhomogenization is that the effects of rapid oscillations upon the solution is averagedout.

As an application we show that for particular choices of aε and bε it is possibleto analyze the effects of multiscale surface roughness in some interesting Newtonianand non-Newtonian lubrication models. For example both the stationary incom-pressible Reynolds equation and the Reynolds-type equation derived in [13] basedon the Rabinowitsch constitutive relation, belong to this category. It is well knownthat the surface micro topography has a significant effect on the hydrodynamicperformance in thin film lubrication. Machined surfaces are never perfectly smoothbecause of defects in the manufacturing process. Since smoothening of the surfacesin contact in a fluid film bearing may lead to a decrease in performance, bearing

Key words and phrases. Reiterated homogenization, monotone operator, multiscale conver-

gence, three-scale convergence, hydrodynamic lubrication, non-Newtonian lubrication, Reynoldsequation, Reynolds-type equation, surface roughness, p-Laplace.

1

2 A. ALMQVIST, E. K. ESSEL, J. FABRICIUS, AND P. WALL

designers are also considering artifically textured surfaces. The effects of surfaceroughness in various lubrication regimes have been studied with homogenizationtechniques in numerous works, e.g. [5, 6, 7, 8, 9, 14, 22]. The usual assumption isthat the roughness is periodic with characteristic wavelength ε. The main resultof this paper makes it possible to study surface roughness with two distinguishablewavelengths, say ε and ε2, in both linear and nonlinear lubrication models.

Reiterated homogenization of −div aε(x,∇uε) = f with non-oscillating f hasbeen studied in [21] by the periodic unfolding method. Reiterated homogenizationof −div aε(x,∇uε) = fε, where fε converges strongly in W−1,q(Ω), has also beenstudied in [16, 18]. This also differs from the present case in that div bε, in general,does not converge strongly in W−1,q(Ω).

2. Preliminaries and notation

Suppose p, α, β, λ and θ are constants that obey

(2) 1 < p <∞, 0 < α ≤ min1, p− 1, max2, p ≤ β <∞, λ, θ > 0.

A function f : RN → RN is said to belong to the class Mpα,β(λ, θ) provided the

following conditions are satisfied for any ξ, η ∈ RN .

f(0) = 0,(3a)

|f(ξ)− f(η)| ≤ λ(1 + |ξ|+ |η|)p−1−α |ξ − η|α ,(3b) [f(ξ)− f(η)

]· (ξ − η) ≥ θ

|ξ − η|β

(1 + |ξ|+ |η|)β−p.(3c)

Moreover, M(λ, θ) denotes the set of all linear mappings f : RN → RN such thatf ∈M2

1,2(λ, θ). The function aε : Ω× RN → RN is assumed to be of the form

(4) aε(x, ξ) = a(x,x

ε,x

ε2, ξ)

(x ∈ Ω, ξ ∈ RN )

where a : Ω× RN × RN × RN → RN is a function of Caratheodory type such that• a is periodic (with respect to the unit cell Y = Z = [0, 1]N ) in both the

second and the third argument,• there exists constants p, α, β, λ and θ satisfying (2), such that a(x, y, z, ·) ∈Mp

α,β(λ, θ) for a.e. (x, y, z) ∈ Ω× [0, 1]N × [0, 1]N .

As a consequence of (3a–c), for each ξ ∈ RN , aε satisfies the growth condition

(5) |aε(·, ξ)| ≤ λ(1 + |ξ|)p−1 a.e. in Ω

and the coercivity condition

(6) aε(·, ξ) · ξ ≥ 2p−βθ ×

|ξ|β if |ξ| ≤ 1|ξ|p otherwise

a.e. in Ω.

It is further assumed that bε is of the form

bε(x) = b(x,x

ε,x

ε2

)with b ∈ Lq(Ω;Cper(Y × Z)).

A weak solution of (1) is defined as an element uε of W 1,p0 (Ω) satisfying

(7)∫

Ω

aε(x,∇uε) · ∇φdx =∫

Ω

bε · ∇φdx

REITERATED HOMOGENIZATION 3

for all φ ∈W 1,p0 (Ω). Let Aε : W 1,p

0 (Ω) →W−1,q(Ω) be defined by

〈Aε(u), φ〉 =∫

Ω

aε(x,∇u) · ∇φdx (u, φ ∈W 1,p0 (Ω)).

By (3b) and Holder’s inequality , we obtain

‖Aε(u)−Aε(v)‖W−1,q(Ω) ≤ λ∥∥1 + |∇u|+ |∇v|

∥∥p−1−α

Lp(Ω)‖u− v‖α

W 1,p0 (Ω) .

Hence Aε is continuous. Owing to (3c) it can be shown that Aε is strictly monotone,i.e.

〈Aε(u)−Aε(v), u− v〉 ≥ 0with equality if and only if u = v. Utilizing (6) and Poincare’s inequality yields

(8) 〈Aε(u), u〉 =∫

Ω

aε(x,∇u) · ∇u dx ≥ const ‖u‖p

W 1,p0 (Ω)

− 2p−βθmeas (Ω),

implying that Aε is coercive. Thus the hypotheses of the Browder–Minty theorem(see e.g. [23] p. 557) are verified, and we conclude that for each ε > 0, there existsa unique uε that solves (7). Moreover, putting φ = uε in (7) and utilizing (8) andYoung’s inequality we obtain

(9) ‖uε‖p ≤ const(‖div bε‖q

W−1,q(Ω) + 1).

Since the sequence div bε is weak∗ convergent, and hence bounded, this shows thatthe sequence of solutions uε is bounded in W 1,p

0 (Ω).

3. Three-scale convergence

In 1989 Nguetseng [19] introduced a method for analyzing homogenization prob-lems that was later further developed by Allaire [1] and called two-scale convergence.Two-scale convergence in the setting of Lebesgues spaces Lp with 1 < p < ∞ isdescribed in [20]. An advantage of the two-scale convergence method is that it isdesigned to avoid many of the classical difficulties encountered in the homogeniza-tion process, such as passing to the limit in the product of two weakly convergentsequences, reducing it to an almost trivial process. A limitation of the method isthat one is more or less restricted to the periodic case.

Homogenization of problems with n microscopic scales, is for n > 1 referred toas reiterated homogenization, see e.g. [10] and [16]. Two-scale convergence (onemicroscopic scale) has been generalized to (n + 1)-scale convergence or multiscaleconvergence (n microscopic scales) by Allaire and Briane [2]. As pointed out bythe authors of that work, the n-scale case is more delicate and for the special casen = 1 the proofs are sometimes much simpler compared to the general case. Theresults of [2] being restricted to L2(Ω), it has hitherto not been clear whether amultiscale theory is possible for Lp(Ω). Nevertheless, many authors have claimedsuch results, see e.g. Theorem 3.8 in [4] or Theorem 1.7 in [12], without providingany complete proof. An additional result of this paper is that we develop such atheory for the case of two microscopic scales (n = 2) and 1 < p <∞. In particularwe give a new proof of Theorem 1.2 (see also Theorem 2.6) in [2]. The assumptionthat the micro scale is the square of the meso scale corresponds to the notion ofwell-separated scales defined in [2].

With the appropriate notion of three-scale convergence combined with an addi-tional result concerning three-scale convergence and monotonicity, homogenizationof (7) becomes a rather short story. We mention that for the case that aε is a


matrix, the homogenized equation obtained by three-scale convergence coincideswith that of [3], which was obtained by the asymptotic expansion method and canthus be seen as a rigorous justification of this method.

Definition 3.1. A bounded sequence uε (ε > 0) in Lp(Ω) is said to three-scaleconverge to an element u of Lp(Ω× Y × Z) provided

(10) limε→0

∫Ω

uε(x)φ(x,x

ε,x

ε2

)dx =

∫∫∫Ω Y Z

u(x, y, z)φ(x, y, z) dz dy dx

for every test function φ of the form φ(x, y, z) = ϕ(x)ψ(y)σ(z), where ϕ ∈ C(Ω),ψ ∈ Cper(Y ) and σ ∈ Cper(Z).

We note that we have an equivalent definition of two-scale convergence if wereplace the set of test functions by D(Ω;C∞per(Y × Z)), i.e. the space that consistsof all functions Ω → C∞per(Y ×Z) such that for any x ∈ Ω, u(x, ·) ∈ C∞per(Y ×Z) andthe mapping Ω 3 x 7→ u(x, ·) ∈ C∞per(Y ×Z) is infinitely differentiable (in the senseof Frechet) with compact support in Ω. For the more general case of periodicitythat Y and Z are parallelograms in RN , Definition 3.1 must be modified by dividingthe right hand side of (10) with meas(Y )meas(Z).

As a direct consequence of the definition of three-scale convergence it is true thatany three-scale convergent sequence uε is also weakly convergent in Lp(Ω). Moreprecisely,

uε → v weakly, v(x) =∫∫Y Z

u(x, y, z) dz dy,

whenever uε → u three-scale. This follows from taking ψ = σ = 1 in (10).We state below the three most important theorems in the theory of multiscale

convergence.

Theorem 3.2 (Three-scale compactness). For any bounded sequence uε in Lp(Ω),there exists a subsequence that three-scale converges weakly.

Proof. The proof is very similar to the two-scale case, see Theorem 7 in [20].

To prove the next important theorem we need a lemma concerning a specialtype of convergence for periodic functions. For the sake of brevity, the proof ispostponed to the end of the paper.

Lemma 3.3. Assume ψ1 ∈ C∞c (Ω), ψ2 ∈ C∞per(Y ) and f ∈ Lpper(Z) satisfies∫

Z

f dz = 0.

Then the sequence of functions fε, defined for a.e. x ∈ Ω by

fε(x) =1ε2ψ1(x)ψ2

(xε

)f( xε2

),

converges weak∗ to 0 in W−1,p(Ω). In particular

supε>0

‖fε‖W−1,p(Ω) <∞.

Theorem 3.4 (Three-scale convergence of the gradient). Suppose that uε is asequence in W 1,p

0 (Ω) such that(1) uε → u weakly in W 1,p

0 (Ω),


(2) ∇uε → ξ ∈ Lp(Ω× Y × Z; RN ) three-scale.

Then uε → u three-scale and there exists u1 ∈ Lp(Ω;W 1,pper(Y )) and u2 ∈ Lp(Ω ×

Y ;W 1,pper(Z)) such that

ξ(x, y, z) = ∇u(x) +∇yu1(x, y) +∇zu2(x, y, z).

Proof.Step 1. Since uε is a bounded sequence in Lp(Ω) it is possible to extract a three-scale convergent subsequence (still denoted by uε), say uε → u′ three-scale. Byintegration by parts we obtain the identity

(11)∫

Ω

∇uε(x) · Φ(x,x

ε,x

ε2

)dx

= −∫

Ω

uε(x)(divx + ε−1divy + ε−2divz)Φ(x,x

ε,x

ε2

)dx

for all Φ ∈ D(Ω;C∞per(Y × Z; RN )). Multiplying (11) with ε2 and passing to thesubsequential three-scale limit u′ of uε we obtain

−∫∫∫Ω Y Z

u′(x, y, z)divzΦ(x, y, z) dz dy dx = 0.

It follows that u′ does not depend on z. Similarly, taking Φ in (11) independent ofz and multiplying by ε yields

−∫∫Ω Y

u′(x, y)divyΦ(x, y) dy dx = 0

and we conclude that u′ does not depend on y either. The three-scale convergenceimplies uε → u′ weakly in Lp(Ω) (always for the same subsequence). Since theembedding of W 1,p(Ω) in Lp(Ω) is compact we also have that uε → u strongly inLp(Ω) (for the whole sequence). It follows that u′ = u and consequently the wholesequence uε and not just a subsequence three-scale converges to u.

Step 2. From two-scale theory, Theorem 13 in [20], we know that

ξ(x, y) = ∇u(x) +∇yu1(x, y) where ξ =∫

Z

ξ dz.

This follows from the fact that ∇uε → ξ two-scale. Next we project ξ − ξ onto thespace of z-gradients. That is, define

Lppot(Z) =

∇zv : v ∈ Lp(Ω× Y ;W 1,p

per(Z)).

Because of the Poincare–Wirtinger inequality, Lppot(Z) is a closed subspace of

Lp(Ω × Y × Z; RN ) and the latter being uniformly convex there exists a ∇zu2 ∈Lp

pot(Z) which minimizes the distance from ξ − ξ to Lppot(Z), i.e.

(12)∥∥ξ − ξ −∇zu2

∥∥ = minΦ∈Lp

pot(Z)

∥∥ξ − ξ − Φ∥∥ .

Let η be defined by

ξ(x, y, z) = ∇u(x) +∇yu1(x, y) +∇zu2(x, y, z) + η(x, y, z).


Computing the first variation of the minimization problem (12) we obtain∫∫∫Ω Y Z

|η|p−2η · Φ dz dy dx = 0

for all Φ of the formΦ(x, y, z) = φ(x)ψ(y)∇σ(z)

with φ ∈ C∞c (Ω), ψ ∈ C∞per(Y ) and σ ∈W 1,pper(Z). It follows that for a.e. x ∈ Ω and

a.e. y ∈ Y , ∫Z

|η|p−2η(x, y, z) · ∇σ(z) dz = 0 for all σ ∈W 1,p

per(Z).

To summarize we have

(13)∫

Z

η dz = 0 and divz

(|η|p−2

η)

= 0.

Step 3. We show that η = 0. To prove this, we take test functions in (11) of theform

(14) Φ(x, y, z) = φ(x)ψ(y)σ(z)

with φ ∈ C∞c (Ω), ψ ∈ C∞per(Y ) and σ ∈ C∞per(Z; RN ) satisfying

div σ = 0,(15a) ∫Z

σ dz = 0.(15b)

For such Φ (11) reduces to∫Ω

∇uε · Φ dx = −∫

Ω

uε

(ψ(xε

)∇φ(x) · σ

( xε2

)+ ε−1φ(x)∇ψ

(xε

)· σ( xε2

))dx.

Letting ε→ 0 we obtain

(16)∫∫∫Ω Y Z

ξ · Φ dz dy dx = limε→0

ε

∫Ω

ϕεuε dx

whereϕε(x) =

1ε2φ(x)∇ψ

(xε

)· σ( xε2

).

Applying Lemma 3.3, we conlude that ϕε is bounded in W−1,q(Ω). Since uε is alsobounded in W 1,p

0 (Ω), it holds that∣∣∣∣∫Ω

ϕεuε dx

∣∣∣∣ = |〈ϕε, uε〉| ≤ ‖ϕε‖W−1,q(Ω) ‖uε‖W 1,p(Ω) ≤ const.

Thus (16) actually says that∫∫∫Ω Y Z

ξ · Φ dz dy dx = 0 implying∫∫∫Ω Y Z

η · Φ dz dy = 0

for all Φ having the special form (14) and satisfying conditions (15), but in view of(13) it is clear that we can omit condition (15b). Thus∫∫∫

Ω Y Z

(η · σ(z)

)φ(x)ψ(y) dz dy dx = 0,


where σ is assumed to satisfy only divzσ = 0. It follows by Fubini’s theorem,density etc. that for a.e. (x, y) ∈ Ω× Y∫

Z

η(x, y, z) · σ(z) dz = 0

for all σ ∈ Lqper(Z) such that div σ = 0. Thus, for fixed x and y, we can take σ(z) =

|η|p−2η(x, y, z). Hence η = 0 a.e. in Ω× Y × Z and ξ = ∇u+∇yu1 +∇zu2.

The following theorem has been referred to as the “fundamental theorem ofthree-scale convergence and monotonicity”. A two-scale version of the statementcan be found in [17], Theorem 14. Our proof is very similar, but we include it herefor the sake of completeness.

Theorem 3.5 (Three-scale convergence and monotonicity). Assume aε as in (4)and let vε be a bounded sequence in Lp(Ω; RN ) such that

vε → v three-scale and aε(x, vε) → ζ three-scale,

for v, ζ ∈ Lp(Ω× Y × Z). Then

(17) lim infε→0

∫Ω

aε(x, vε) · vε dx ≥∫∫∫Ω Y Z

ζ · v dx dy dz

and if equality holds, then ζ = a(·, v).

Proof. Let Φ(x, y, z) be a linear combination of vector fields of the form (x, y, z) 7→φ(x)ψ(y)σ(z)ν where φ ∈ C∞c (Ω), ψ ∈ C∞per(Y ), σ ∈ C∞per(Z) and ν ∈ SN−1 (theunit sphere). By monotonicity∫

Ω

(aε(x, vε)− aε(x,Φε)

)· (vε − Φε) dx ≥ 0.

whereΦε(x) = Φ

(x,x

ε,x

ε2

).

Rearranging the terms yields∫Ω

aε(x, vε) · vε dx ≥∫

Ω

aε(x, vε) · Φε + aε(x,Φε) · (vε − Φε) dx.

Since the limit, as ε→ 0, of the right hand side of the above inequality exists andis equal to ∫∫∫

Ω Y Z

ζ · Φ + a(x, y, z,Φ) · (v − Φ) dx dy dz

we have

(18) lim infε→0

∫Ω

aε · vε dx ≥∫∫∫Ω Y Z

ζ · Φ + a(x, y, z,Φ) · (v − Φ) dx dy dz

and by density and continuity this also holds for all Φ in Lp(Ω× Y ×Z). Thus, weestablish (17) by taking Φ = v.

Suppose now that equality holds in (17). For some w ∈ Lp(Ω× Y ×Z; RN ) andt ∈ R, choose Φ = v + tw in (18) to obtain

0 ≥ t

∫∫∫Ω Y Z

(ζ − a(x, y, z, v + tw)

)· w dxdy dz.


Dividing by t and using the continuity of a we let t→ 0±, thus obtaining∫∫∫Ω Y Z

(ζ − a(x, y, z, v)

)· w dxdy dz = 0

for all w ∈ Lp(Ω × Y × Z; RN ). Hence ζ(x, y, z) = a(x, y, z, v(x, y, z)) almosteverywhere.

4. A three-scale homogenization procedure

Based on Theorems 3.2, 3.4 and 3.5, we outline a homogenization procedure forthe problem (1).

In view of estimate (9) and the following remark, the sequence of solutions uε

to (7) is bounded in W 1,p0 (Ω). Applying Theorems 3.2 and 3.4 we can find u ∈

W 1,p0 (Ω), u1 ∈ Lp(Ω;W 1,p

per(Y )), u2 ∈ Lp(Ω×Y ;W 1,pper(Z)) and ζ ∈ Lq(Ω×Y ×Z)N

such that up to a subsequence

(1) uε → u three-scale,(2) ∇uε → ∇u+∇yu1 +∇zu2 three-scale,(3) aε(x,∇uε) → ζ three-scale.

Passing to the limit in the weak formulation (7) gives∫∫∫Ω Y Z

ζ · ∇φdz dy dx =∫∫∫Ω Y Z

b · ∇φdz dy dx

Let the testfunction φ in (7) be φ(x) = εφ1(x)w1(x/ε), where φ1 ∈ C∞c (Ω), w1 ∈C∞per(Y ). Then∫

Ω

aε(x,∇uε) · (εw1∇φ1 + φ1∇w1) dx =∫

Ω

bε · (εw1∇φ1 + φ1∇w1) dx.

In the limit as ε→ 0 we obtain∫∫∫Ω Y Z

ζ · φ1(x)∇w1(y) dz dy dx =∫∫∫Ω Y Z

b · φ1(x)∇w1(y) dz dy dx.

Taking as testfunction φ(x) = ε2φ1(x)φ2(x/ε)w2(x/ε2), where φ1 ∈ C∞c (Ω), φ2 ∈C∞per(Y ) and w2 ∈ C∞per(Z), yields∫

Ω

aε(x,∇uε) ·(ε2w2φ2∇φ1 + εw2φ1∇φ2

(xε

)+ φ1φ2∇w2

( xε2

))dx

=∫

Ω

bε ·(ε2w2φ2∇φ1 + εw2φ1∇φ2

(xε

)+ φ1φ2∇w2

( xε2

))dx.

In the limit∫∫∫Ω Y Z

ζ · φ1(x)φ2(y)∇w2(z) dz dy dx =∫∫∫Ω Y Z

b · φ1(x)φ2(y)∇w2(z) dz dy dx.


By density it follows that ζ satisifies

(19)∫∫∫Ω Y Z

ζ · (∇φ+∇yφ1 +∇zφ2) dz dy dx

=∫∫∫Ω Y Z

b · (∇φ+∇yφ1 +∇zφ2) dz dy dx

for all φ ∈ W 1,p0 (Ω), φ1 ∈ Lp(Ω;W 1,p

per(Y )), φ2 ∈ Lp(Ω × Y ;W 1,pper(Z)). Let us now

characterize ζ. Choosing φ = u, φ1 = u1 and φ2 = u2 in the identity (19) gives

(20)∫∫∫Ω Y Z

ζ · (∇u+∇yu1 +∇zu2) dz dy dx

=∫∫∫Ω Y Z

b · (∇u+∇yu1 +∇zu2) dz dy dx.

Taking φ = uε in (7) yields

limε→0

∫Ω

aε(x,∇uε) · ∇uε dx = limε→0

∫Ω

bε · ∇uε dx

=∫∫∫Ω Y Z

b · (∇u+∇yu1 +∇zu2) dz dy dx.(21)

Combining (20) and (21), we see that

limε→0

∫Ω

aε(x,∇uε) · ∇uε dx =∫∫∫Ω Y Z

ζ · (∇u+∇yu1 +∇zu2) dz dy dx.

Appealing to the fundamental theorem of three-scale convergence and monotonicity,Theorem 3.5 we conclude that

ζ = a(·,∇u+∇yu1 +∇zu2).

Hence, u, u1 and u2 satisfy the following homogenized variational system

(22)∫∫∫Ω Y Z

a(x, y, z,∇u+∇yu1 +∇zu2) · (∇φ+∇yφ1 +∇zφ2) dz dy dx

=∫∫∫Ω Y Z

b · (∇φ+∇yφ1 +∇zφ2) dz dy dx

for all φ ∈ W 1,p0 (Ω), φ1 ∈ Lp(Ω;W 1,p

per(Y )), φ2 ∈ Lp(Ω × Y ;W 1,pper(Z)). We have

obtained the following partial homogenization result.

Theorem 4.1. For ε > 0, let uε ∈W 1,p0 (Ω) denote the solution of∫

Ω

aε(x,∇uε) · ∇φdx =∫

Ω

bε · ∇φdx ∀φ ∈W 1,p0 (Ω).

Then there exists a subsequence of solutions uεjsuch that uεj

→ u weakly and∇uεj

→ ∇u + ∇yu1 + ∇zu2 three-scale, as εj → 0. In addition u ∈ W 1,p(Ω),u1 ∈ Lp(Ω;W 1,p

per(Y )) and u2 ∈ Lp(Ω× Y ;W 1,pper(Z)) solve the system (22).


A priori, it is not clear that u, u1 and u2 are uniquely determined by the factthat they solve the system (22). To make the analysis complete it remains to provethis. To this end let a∗ be defined by

(23) a∗(x, y, ξ) =∫

Z

a(x, y, z, ξ +∇ψ∗) dz (x ∈ Ω, y, ξ ∈ RN ),

where ψ∗ ∈W 1,pper(Z) is a solution of the variational problem

(24)∫

Z

a(x, y, z, ξ +∇ψ∗) · ∇ψ dz =∫

Z

b(x, y, z) · ∇ψ dz ∀ψ ∈W 1,pper(Z),

and define a∗∗ as

(25) a∗∗(x, ξ) =∫

Y

a∗(x, y, ξ +∇ψ∗∗) dy (x ∈ Ω, ξ ∈ RN ),

where ψ∗∗ ∈W 1,pper(Y ) solves

(26)∫

Y

a∗(x, y, ξ +∇ψ∗∗) · ∇ψ dy =∫

Y

b(x, y)∗ · ∇ψ dy ∀ψ ∈W 1,pper(Y ),

where b∗(x, y) =∫

Zb(x, y, z) dz. Since a(x, y, z, ·) ∈ Mp

α,β(λ, θ) it follows by theBrowder–Minty theorem, as explained above for (7), that for a.e. (x, y) ∈ Ω × Ythere exists a solution of (24) that is unique up to an additive constant. Thusa∗ is well defined, however, we can not say the same for a∗∗ unless we know that(26) has a solution that is unique (up to a constant). Assume for the moment thatexistence and uniqueness hold true for (26). Next we show how this can be utilizedto characterize the homogenized solution u from (22).

Taking φ = φ1 = 0 and φ2(x, y, z) = ϕ1(x)ϕ2(y)ψ(z), ϕ1 ∈ C∞c (Ω), ϕ2 ∈C∞per(Y ) and ψ ∈ W 1,p

per(Z), in (22) implies that for a.e. (x, y) ∈ Ω × Y , u2 ∈Lp(Ω× Y ;W 1,p

per(Z)) satisfies

(27)∫

Z

a(x, y, z,∇u(x) +∇yu1(x, y) +∇zu2(x, y, z)

)· ∇ψ dz

=∫

Z

b(x, y, z) · ∇ψ dz.

Consequently, by the definition of a∗,

(28) a∗(x, y,∇u(x) +∇yu1(x, y)

)=∫

Z


)dz.

Similarly, by taking φ = φ2 = 0 in (22) we obtain that for a.e. x ∈ Ω, u1 ∈Lp(Ω;W 1,p

per(Y )) satisfies∫Y

a∗(x, y,∇u(x) +∇yu1(x, y)

)· ∇ψ dy =

∫Y

b∗(x, y) · ∇ψ dy

for all ψ ∈W 1,pper(Y ). Hence

(29) a∗∗(x,∇u(x)

)=∫

Y

a∗(x, y,∇u(x) +∇yu1(x, y)

)dy

=∫∫Y Z


)dz dy.


Finally, setting φ1 = φ2 = 0 in (22) and taking (23) and (25) into account, wesee that u ∈W 1,p

0 (Ω) satisfies the variational identity

(30)∫

Ω

a∗∗(x,∇u) · ∇φdx =∫

Ω

b∗∗ · ∇φdx ∀φ ∈W 1,p0 (Ω),

where

(31) b∗∗(x) =∫∫Y Z

b(x, y, z) dz dy.

We would like to prove that (30) has a unique solution u and for this we need someproperties of a∗ and a∗∗.

Notation. In view of (24), any vector field ξ : Ω × Y → RN induces a vector fieldξ∗ : Ω× Y × Z → RN defined by

ξ∗(x, y, z) = ξ(x, y) +∇ψ∗(z),

where ψ∗ is a solution of (24). Moreover, if ξ is a vector field on Ω, ξ∗∗ is definedas

ξ∗∗(x, y, z) = ξ(x) +∇ψ∗∗(y) +∇ψ∗(z),where ψ∗∗ solves (26) and ψ∗ is a solution of∫

Z

a(x, y, z, ξ(x) +∇ψ∗∗(y) +∇ψ∗

)· ∇ψ dz =

∫Z

b(x, y, z) · ∇ψ dz ∀ψ ∈W 1,pper(Z).

.

Theorem 4.2. The function a∗, defined by (23), satisfies certain continuity andmonotonicity conditions so that existence and uniqueness of (26) is guaranteed fora.e. x ∈ Ω. Thus the function a∗∗, in (25), is well defined. Moreover a∗∗ issufficiently continuous and monotone in ξ so that existence and uniqueness of (30)follows.

Sketch of proof. Set

α∗ =α

β − αand λ∗ =

(λ

θ

)α∗

λ.

Utilizing the monotonicity and continuity of a it follows by some straight-forwardcalculations, for the details of which we refer to [16] (Propopsition 3.1) and [11](Lemma 3.4), that for every ξ, η ∈ RN

|a∗(·, ξ)− a∗(·, η)| ≤ λ∗∥∥1 + |ξ∗|+ |η∗|

∥∥p−1−α∗

Lp(Z)|ξ − η|α

∗(32) [

a∗(·, ξ)− a∗(·, η)]· (ξ − η) ≥ θ

|ξ − η|β∥∥1 + |ξ∗|+ |η∗|∥∥β−p

Lp(Z)

(33)

holds a.e. in Ω× Y , and

|a∗∗(·, ξ)− a∗∗(·, η)| ≤ λ∗∥∥1 + |ξ∗∗|+ |η∗∗|

∥∥p−1−α∗

Lp(Y×Z)|ξ − η|α

∗(34) [

a∗∗(·, ξ)− a∗∗(·, η)]· (ξ − η) ≥ θ

|ξ − η|β∥∥1 + |ξ∗∗|+ |η∗∗|∥∥β−p

Lp(Y×Z)

(35)

holds a.e. in Ω.


Next we show that there exists a c(x) such that for any vector field ξ ∈ Lp(Y ; RN )it holds that

(36) ‖ξ∗‖Lp(Y×Z) ≤ c(x)(1 + ‖ξ‖Lp(Y )

)with c(x) <∞ for a.e. x ∈ Ω.

First note the lower bound

‖ξ∗‖Lp(Y×Z) ≥ ‖ξ‖Lp(Y ) .

Indeed

‖ξ‖pLp(Y ) =

∫Y

∣∣∣∣∫Z

ξ∗ dz

∣∣∣∣p dy ≤ ∫∫ |ξ∗|p dz dy.

We now seek to establish an upper bound of ‖ξ∗‖Lp(Y×Z). Without loss of generalitywe may assume ‖ξ∗‖Lp(Y×Z) ≥ 1.

On the one hand (3c), Holder’s inequality and the triangle inequality gives∫∫a(x, y, z, ξ∗) · ξ∗ dz dy ≥ θ

∫∫|ξ∗|β

(1 + |ξ∗|)β−pdz dy

≥ θ

2β−p‖ξ∗‖p

Lp(Y×Z) .

(37)

On the other hand (5) and (26) implies

(38)∫∫a(x, y, z, ξ∗) · ξ∗ dz dy =

∫∫a(x, y, z, ξ∗) · ξ dz dy +

∫∫b · (ξ∗ − ξ) dz dy

≤ λ

∫∫|ξ| (1 + |ξ∗|)p−1dz dy + ‖b‖Lq(Y×Z) ‖ξ

∗ − ξ‖Lp(Y×Z)

≤ 2p−1λ ‖ξ‖Lp(Y ) ‖ξ∗‖p−1

Lp(Y×Z) + 2 ‖b‖Lq(Y×Z) ‖ξ∗‖Lp(Y×Z) .

Combining (37) and (38) we see that

(39) ‖ξ∗‖Lp(Y×Z) ≤ const(‖ξ‖Lp(Y ) + ‖b‖Lq(Y×Z)

)Since b ∈ Lq(Ω;Cper(Y × Z)), we have

‖b(x, ·)‖Lq(Y×Z) ≤ ‖b(x, ·)‖qC(Y×Z) <∞ for a.e. x ∈ Ω.

This establishes (36).By estimating ∫∫∫

a(x, y, z, ξ∗∗) · ξ∗∗ dz dy dx

similarly, one can show that there exists a constant c that depends on p, λ, θ, βand ‖b‖Lq(Ω;C(Y×Z)) such that for any ξ ∈ Lp(Ω; RN )

(40) ‖ξ∗∗‖Lp(Ω×Y×Z) ≤ c(1 + ‖ξ‖Lp(Ω)

)

Summing up we have the following homogenization result.

Theorem 4.3. Let a∗, a∗∗ and b∗∗ be defined by (23), (25) and (31) respectively.Both a∗ and a∗∗ are well defined. For ε > 0, let uε denote the solution of (1).


Then the whole sequence uε converges weakly to u as ε→ 0, where u is the uniqueweak solution of the homogenized problem

(41)div a∗∗(x,∇u) = div b∗∗ in Ω

u = 0 on ∂Ω.

5. The linear case

Let us consider the special case that a(x, y, z, ·) ∈ M(λ, θ). Since a is assumedto be linear we write

a(x, y, z, ξ) = a(x, y, z)ξ and aij(x, y, z)1≤i,j≤N

denotes the corresponding matrix. Note that conditions (3b) and (3c) are satisfiedif aij ∈ L∞(Ω;Cper(Y × Z) and aij is (uniformly) positive definite, i.e.

max1≤i,j≤N

‖aij‖L∞(Ω;Cper(Y×Z)) ≤ λ and∑1≤i,j≤N

aij(x, y, z)ξjξi ≥ θ |ξ|2

for all ξ ∈ RN and a.e. (x, y, z) ∈ Ω × Y × Z. The equation corresponding to (1)then becomes

div(aε∇uε) = div bε in Ω.In the linear situation the analysis is essentialy simplified in the sense that one

only has to solveN+1 local problems corresponding to the z-scale and anotherN+1problems corresponding to the y-scale, instead of infinitely many local problems(one for each ξ), to obtain the homogenized equation (41). Let eii=1,...,N denotethe standard basis in RN . Due to the linearity of a, a solution ψ∗ξ of (24), i.e. asolution of∫

Z

a(x, y, z)(ξ +∇ψ∗ξ ) · ∇ψ dz =∫

Z

b(x, y, z) · ∇ψ dz ∀ψ ∈W 1,pper(Z)

can be written in the form

ψ∗ξ = ψ∗0 +N∑

i=1

ξiχ∗i ,

where ψ∗0 and χ∗i are solutions of

(42a)∫

Z

a(x, y, z)∇ψ∗0 · ∇ψ dz =∫

Z

b(x, y, z) · ∇ψ dz ∀ψ ∈W 1,pper(Z)

and

(42b)∫

Z

a(x, y, z)(ei +∇χ∗i ) · ∇ψ dz = 0 ∀ψ ∈W 1,pper(Z) (i = 1, . . . , N).

Let a1(x, y) be the matrix and b1(x, y) the vector defined by

a1(x, y)ei =∫

Z

a(x, y, z)(ei +∇χ∗i ) dz,(43a)

b1(x, y) =∫

Z

b(x, y, z)− a(x, y, z)∇ψ∗0 dz.(43b)

Then, ψ∗∗ξ solving (26) is equivalent to ψ∗∗ξ solving∫Y

a1(x, y)(ξ +∇ψ∗∗ξ ) · ∇ψ dy =∫

Y

b1(x, y) · ∇ψ dy ∀ψ ∈W 1,pper(Y ).


By linearity, the solution ψ∗∗ξ can be written

ψ∗∗ξ = ψ∗∗0 +N∑

i=1

ξiχ∗∗i ,

where ψ∗∗0 and χ∗∗i are solutions of

(44a)∫

Y

a1(x, y)∇ψ∗∗0 · ∇ψ dy =∫

Y

b1(x, y) · ∇ψ dy ∀ψ ∈W 1,pper(Y )

and

(44b)∫

Y

a1(x, y)(ei +∇χ∗∗i ) · ∇ψ dy = 0 ∀ψ ∈W 1,pper(Y ) (i = 1, . . . , N).

Let a0(x) be the matrix and b0(x) the vector defined by

a0(x)ei =∫

Y

a1(x, y)(ei +∇χ∗∗i ) dy,(45a)

b0(x) =∫

Y

b1(x, y)− a1(x, y)∇ψ∗∗0 dy.(45b)

Summing up we have the following homogenization algorithm:(1) Solve the N + 1 local problems (42) on the z-scale and use these solutions

to compute the matrix a1(x, y) and the vector b1(x, y) in (43).(2) Solve the N + 1 local problems (44) on the y-scale and use these solutions

to compute the matrix a0(x) and the vector b0(x) in (45).(3) Solve the homogenized equation

(46)div(a0∇u) = div b0 in Ω

u = 0 on ∂Ω.

6. Application to hydrodynamic lubrication

The Reynolds equation is a two-dimensional model that describes the flow in athin film of viscous fluid (lubricant) that is enclosed between two rigid surfaces inrelative motion. Reynolds equation is used by engineers to compute the pressuredistribution in various situations of hydrodynamic lubrication, e.g. slider bearingsor journal bearings. A simple form of Reynolds equation reads

(47)∂

∂x1

(h3

12µ∂p

∂x1

)+

∂

∂x2

(h3

12µ∂p

∂x2

)=v

2∂h

∂x1in Ω,

where p is the unknown pressure distribution, Ω is the “bearing domain”, h : Ω → Rdenotes the thickness of the film, µ is the lubricant viscosity (taken as constant) andv is the constant speed of the moving surface (the other remains fixed). To studythe influence of multiscale surface roughness with two characteristic wavelengths,ε and ε2, upon the pressure solution we make the ansatz that the film thicknessfunction hε is given by

hε(x) = h(x,x

ε,x

ε2

), where h : Ω× RN × RN → R

is assumed to be continuous, [0, 1]N -periodic in the second and third argumentsand satisfy θ ≤ h3 ≤ λ. Assuming the boundary condition pε = 0 on ∂Ω, we can


utilize the homogenization result (46) with

aε(x) =(hε(x)3 0

0 hε(x)3

), bε(x) = (6µvhε, 0) and uε = pε.

The Reynolds equation (47) may be a good approximation for Newtonian fluids,but it does not take into account the particular behaviour of non-Newtonian fluids.Attempts have therefore been made to modify (47) so as to capture non-Newtonianeffects. For example He [13] has suggested the following Reynolds-type equationfor one-dimensional incompressible non-Newtonian lubrication

(48)d

dx

(h3

12µ0

dp

dx+kh5

320

(dp

dx

)3)

=v

2dh

dx,

where µ0 is the zero shear rate viscosity and k ≥ 0 denotes a nonlinear factor ac-counting for non-Newtonian effects that comes from the Rabinowitsch constitutiverelation.

To write (48) in the form (1), choose

aε(x, ξ) = h3εξ + k1h

5εξ

3, and bε = k2hε,

where k1 = 3kµ0/80 and k2 = 6µ0v. The homogenization result (41) then applieswith p = 4, α = 1 and β = 4.

7. A convergence result for periodic functions

The objective of this section is to prove Lemma 3.3. First we state a resultpertaining to the image of the div-operator.

Theorem 7.1. For each f ∈ Lpper(Y ) with mean value zero, there exists a vector

field F ∈ Lpper(Y ; RN ) such that

(49)∫

RN

fφ dx = −∫

RN

F · ∇φdx

for all φ ∈ C∞c (RN ). In other words f = divF . Moreover, there exists a constantC > 0 such that

‖F‖Lp(Y ;RN ) ≤ C ‖f‖Lp(Y ) .

Proof. Consider the periodic q-Poisson equation

(50) −∆qu = f in RN

u is Y -periodic,

where ∆q is the q-Laplace operator defined for q ≥ 1 by

∆qu = div(|∇u|q−2∇u

).

By the direct methods of the calculus of variations or the theory for monotoneoperators it can be shown that, for f as above, there exists a weak solution u ∈W 1,q

per(Y ) of (50) that satisfies

(51)∫

Y

|∇u|q−2∇u · ∇v dy =∫

Y

fv dy ∀v ∈W q,1per(Ω).


Thus F = − |∇u|q−2∇u ∈ Lpper(Y ; RN ) satisfies (49). Taking v = u in (51) we

obtain by Holder’s inequality∫Y

|∇u|q dy ≤(∫

Y

|f |p dy) 1

p(∫

Y

|u|q dy) 1

q

.

Noting that |F |p = |∇u|q combined with The Poincare–Wirtinger inequality yields(∫Y

|F |p dy) 1

p

≤ C

(∫Y

|f |p dy) 1

p

.

It is well known (and frequently used in homogenization theory) that for f : Ω×RN → R continuous and Y -periodic in the second argument, the function fε(x) =f(x, x/ε) converges weakly in Lp(Ω) (1 ≤ p <∞) to the function x 7→

∫Yf(x, y) dy.

This result (and its generalizations) is commonly referred to as the “mean valueproperty” or “generalized Riemann–Lebesgues lemma”.

Suppose in addition that for each x ∈ Ω, f satisfies∫

Yf(x, y) dy = 0. Then we

can conclude that fε → 0 weakly in Lp(Ω). The example

limε→0

∫ 1

0

1ε

sin(xε

)φdx = lim

ε→0

∫ 1

0

cos(xε

)φ′ dx = 0 (φ ∈ C∞c (0, 1)),

i.e. Ω = (0, 1) and f(x, y) = sin y, serves to illustrate that the sequence of functions

1ρ(ε)

fε

may also converge, in some sense, provided ρ(ε) → 0 sufficiently slowly as ε → 0.The precise statement, along with the obvious generalization to three scales, aregiven by the following theorems, for the case that

f(x, y) = φ(x)ψ(y) φ ∈ C(Ω), ψ ∈ C∞per(Y ).

For the case p = 2 similar versions of these theorems can be found in [2].

Lemma 7.2. Assume ψ ∈ C∞c (Ω) and f ∈ Lpper(Y ) satisfies∫

Y

f dy = 0.


fε(x) =1εψ(x)f

(xε

),

converges weak∗ to 0 in W−1,p(Ω).

Proof. According to Theorem 7.1 there exists F ∈ Lpper(Y ; RN ) such that∫

RN

fφ dx = −∫

RN

F · ∇φdx ∀φ ∈ C∞c (RN ).

Choosing as test functions x 7→ ψ(εx)φ(εx), where φ ∈ C∞c (RN ) is arbitrary, weobtain by the chain rule and a change of variables∫

RN

fεφdx = −∫

RN

F(xε

)· ∇(ψφ) dx ∀φ ∈ C∞c (RN )


By density we can replace φ with any v ∈W 1,q0 (Ω). Taking the limit, we obtain

limε→0

〈fε, v〉 = −∫

Ω

(∫Y

F dy

)· ∇(ψv) dx = 0.

The strong generalization of Lemma 7.2, by adding the scale ε2, can not beobtained right away. First we need to establish a weaker result.

Lemma 7.3. Assume ψ1 ∈ C∞c (Ω), ψ2 ∈ C∞per(Y ) and f ∈ Lpper(Z) satisfies∫

Z

f dz = 0.


fε(x) =1εψ1(x)ψ2

(xε

)f( xε2

),

converges weak∗ to 0 in W−1,p(Ω).

Proof. There exists F ∈ Lpper(Z; RN ) such that∫

RN

f(x)ψ1(ε2x)ψ2(εx)φ(ε2x) dx = −∫

RN

F (x) · ∇(ψ1(ε2x)ψ2(εx)φ(ε2x)

)dx

for all φ ∈ C∞c (RN ). After a change of variables we obtain∫RN

fεφdx = −ε∫

RN

ψ2

(xε

)F( xε2

)·∇(ψ1φ) dx+

∫RN

(F( xε2

)· ∇ψ2

(xε

))ψ1φdx.

Thus

limε→0

∫RN

fεφdx = limε→0

∫RN

(F( xε2

)· ∇ψ2

(xε

))ψ1φdx

=∫

RN

(∫Z

F (z) dz)·(∫

Y

∇ψ2 dy

)ψ1φdx = 0,

where we used the periodicity of ψ2 in the last equality.

We can now give the postponed proof of Lemma 3.3.

Proof of Lemma 3.3. According to the proof of Lemma 7.3 we have∫RN

fεφdx

= −∫

RN

ψ2

(xε

)F( xε2

)· ∇(ψ1φ) dx+

∫RN

1ε

(F( xε2

)· ∇ψ2

(xε

))ψ1φdx.

By Riemann–Lebesgue lemma, the first term tends to

−∫

RN

(∫Y

ψ2 dy

)(∫Z

F dz

)· ∇(ψ1φ) dx = 0.

Upon noting that the second term can be written as∫RN

1ε

(F( xε2

)− µ

)· ∇ψ2

(xε

)ψ1φdx+

∫Ω

1εµ · ∇ψ2

(xε

)ψ1φdx,


where µ ∈ RN denotes the average of F over Z, we appeal to Lemma 7.3 andLemma 7.2 to conclude that

limε→0

∫RN

fεφdx = 0.

By density we can take φ ∈W 1,q0 (Ω) and the proof is complete.

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surface. Math. Modelling and Numer. Anal., 29(2):199–233, 1995.[15] J.-L. Lions, D. Lukkassen, L.-E. Persson and P. Wall. Reiterated homogenization of monotone

operators. C. R. Acad. Sci. Paris, Ser. I, 330:675–680, 2000.

[16] J.-L. Lions, D. Lukkassen, L.-E. Persson and P. Wall. Reiterated homogenization of nonlinearmonotone operators. Chin. Ann. of Math., 22B(1):1–12, 2001.

[17] D. Lukkassen and P. Wall. Two-scale convergence with respect to measures and homogeniza-tion of monotone operators. J. Funct. Spaces Appl., 3(2):125–161, 2005.

[18] D. Lukkassen, A. Meidell and P. Wall. Multiscale homogenization of monotone operators.

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[20] G. Nguetseng, D. Lukkassen and P. Wall. Two-scale convergence. Int. J. Pure Appl. Math.2(1):35–86, 2002.

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[23] E. Zeidler. Nonlinear functional analysis and its applications II/B. Springer-Verlag,New York Berlin Heidelberg, 1990.


Division of Machine Elements, Lulea University of Technology, SE-971 87 Lulea,Sweden

E-mail address: [email protected]

Department of Mathematics and Statistics, University of Cape Coast, Cape Coast,Ghana

Current address: Department of Mathematics, Lulea University of Technology, SE-971 87Lulea, Sweden


Department of Mathematics, Lulea University of Technology, SE-971 87 Lulea, Swe-den


Department of Mathematics, Lulea University of Technology, SE-971 87 Lulea, Swe-

denE-mail address: [email protected]

mailto:[email protected]




Appendix to A

APPENDIX TO A

JOHN FABRICIUS

Abstract. We develop some elements of analysis on the torus and prove a

regularity result for the p-Poisson equation on the torus.

1. Analysis on the torus

The “N -dimensional torus” TN is defined as the quotient of the groupRN modulothe subgroup of integers ZN , i.e.

TN = R

N/ZN .

The corresponding equivalence relation is: x, y ∈ RN are equivalent provided x−y ∈Z

N . For x ∈ RN , let x denote the corresponding equivalence class in TN . Supposef : TN → R. For x ∈ RN , x → f(x) then defines a periodic function. In this sensewe identify Cper, the space of continuous periodic functions on R

N , with C(TN ),the space of continuous functions on the torus,

Cper C(TN ).

1.1. Integration. Suppose χ is a bounded measurable function on RN with com-pact support such that

(1)∑

k∈ZNχ(x − k) = 1 a.e. in RN .

For f ∈ C(TN ), we define ∫TN

f dx =∫RN

fχ dx

The definition makes sense, because assume χ and χ′ are both bounded measurablefunctions with compact support that satisfy (1). Then∫

RN

fχ dx =∑

k∈ZN

∫RN

f(x)χ(x)χ′(x − k) dx =∑

k∈ZN

∫RN

f(x)χ(x + k)χ′(x) dx

=∫RN

fχ′ dx.

The completion of C(TN ) with respect to Lp-norm is denoted by Lp(TN ). Thanksto the integral we can define the notions of weak derivatives and Sobolev spaces.

The periodic p-Poisson equation is written

(2) −Δpu = f in TN

where Δp is the p-Laplace operator defined for p ≥ 1 by

Δpu = div(|∇u|p−2 ∇u

).

1

2 J. FABRICIUS

The following theorem says that a weak solution of (2) exists, provided f ∈ Lq(TN ),1/p + 1/q = 1 and

∫TN

f dx = 0.

Theorem 1.1. For each f ∈ Lq(TN ) that has zero integral, there exists a u ∈W 1,p(TN ) such that

(3)∫TN

|∇u|p−2 ∇u · ∇v dx =∫TN

fv dx

for all v ∈ W 1,p(T), 1p + 1

q = 1. Moreover, there exists a constant C > 0 such that∫TN

|∇u|p dx ≤ C

∫TN

|f |p dx.

Proof. Consider the minimization problem

I[u] = minv∈W 1,p(T)

I[v]

I[v] =∫TN

|∇v|p − pfv dx.

Let (v) denote the average of v over TN . By using the Young inequality

ab ≤ r−q

q|a|q +

rp

p|b|p ,

with appropriate r > 0, and the Poincare–Wirtinger inequality we obtain

I[v] =∫TN

|∇v|p dx − p

∫TN

f(v − (v)

)dx

≥ 12

∫TN

|∇v|p dx − const∫TN

|f |q dx.

From this estimate we also see that the infimum of I is indeed finite for 1 < p < ∞.It then follows that a minimizing sequence un of I is bounded in W 1,p(T), so wecan extract a subsequence that converges weakly to some element u of W 1,p(T).Since I is weakly lower semicontinuous, u satisfies

I[u] ≤ I[v] ∀v ∈ W 1,p(T).

Having proved the existence of a minimizer u we turn to deriving the correspondingEuler–Lagrange equation. Pick a v ∈ W 1,p(T). For t ∈ R, define i(t) = I[u + vt].Then

i(t) − i(0)t

=1t

(∫TN

|∇u + t∇v|p − pf(u + tv) dx −∫TN

|∇u|p − pfu dx

)=∫TN

|∇u + t∇v|p − |∇u|p

tdx − p

∫TN

fv dx

By the dominated convergence theorem we have

i′(0) = p

∫TN

|∇u|p−2 ∇u · ∇v dx − p

∫TN

fv dx.

Since i(0) ≤ i(t) for all t, it must hold that i′(0) = 0. Hence the minimizer usatisifies

(4)∫TN

|∇u|p−2 ∇u · ∇v =∫TN

fv dx ∀v ∈ W 1,p(T).

APPENDIX TO A 3

Note that u is not unique and that we may assume u satisfies (u) = 0. Bychoosing v = u in (4) we obtain∫

TN

|∇u|p dx =∫TN

fu dx ≤(∫

TN

|f |q dx

)1/q(∫TN

|u|p dx

)1/p

.

Employing the Poincare–Wirtinger inequality yields the estimate(∫TN

|∇u|p dx

)1/q

≤ const(∫

TN

|f |q dx

)1/q

.

Definition 1.2 (Divergence free vector fields). We say that a vector field F ∈L1(TN ;RN ) is “divergence free” or “solenoidal” or div F = 0 if∫

RN

F · ∇φ dx = 0

holds for all φ ∈ C∞c (RN ).

The following result is frequently used in homogenization theory. We give asimple proof of this fact.

Theorem 1.3. A vector field F ∈ L1(TN ;RN ) is divergence free if and only if∫TN

F · ∇φ dx = 0 for all φ ∈ C∞(TN ).

Proof. Choose a ψ ∈ C∞c (RN ), whose integer translates form a partition of unity

of RN , to represent integration on the torus.(=⇒) Assume φ ∈ C∞(TN ). Since φψ ∈ C∞

c (RN ) we have

0 =∫RN

F · ∇(φψ) dx

=∫RN

(F · ∇φ)ψ dx +∑

k∈ZN

∫RN

(F (x) · ∇ψ(x))φ(x)ψ(x − k) dx

=∫RN

(F · ∇φ)ψ dx +∑

k∈ZN

∫RN

(F (x) · ∇ψ(x + k))φ(x)ψ(x) dx

=∫TN

F · ∇φdx + 0.

(⇐=) Suppose now that φ ∈ C∞c (RN ). Then the “periodic push forward” of φ,

is the functionφ(x) =

∑k∈ZN

φ(x + k).

Clearly

0 =∫TN

F · ∇φ dx =∑

k∈ZN

∫RN

(F (x) · ∇φ(x + k))ψ(x) dx

=∑

k∈ZN

∫RN

(F (x) · ∇φ(x))ψ(x − k) dx

=∫RN

F · ∇φdx.

4 J. FABRICIUS

2. Regularity for the p-Poisson equation

Some of the material in this section is covered in [1].

2.1. Some inequalities for vectors.

Lemma 2.1. Assume 1 < p < 2. The inequality(|a|p−2

a − |b|p−2b)· (a − b) > (p − 1)

(|a| + |b|

)p−2 |a − b|2

holds for all a, b ∈ RN , a = 0, b = 0.

Proof. By the fundamental theorem of calculus and the Cauchy–Schwarz inequalitywe have(

|a|p−2a − |b|p−2

b)· (a − b) =

N∑i=1

(ai − bi)∫ 1

0

d

ds

(|ξ|p−2

ξi

∣∣∣ξ=sa+(1−s)b

)ds

=∫ 1

0

|ξ|p−2 |η|2 + (p − 2) |ξ|p−4 |ξ · η|2∣∣∣ξ=sa+(1−s)b

η=(a−b)

ds

≥ (p − 1) |a − b|2∫ 1

0

|sa + (1 − s)b|p−2ds.

By the triangle inequality and the fact that t → tp−2 is decreasing we estimate∫ 1

0

|sa + (1 − s)b|p−2ds ≥

∫ 1

0

(s |a| + (1 − s) |b|

)p−2ds ≥

(|a| + |b|

)p−2.

Lemma 2.2. Assume 1 < p < 2. The inequality

(5)(|a|p−2

a − |b|p−2b)· (a − b) ≥ 1

p

∣∣∣|a| p−22 a − |b|

p−22 b∣∣∣2

holds for all a, b ∈ RN , a = 0, b = 0.

Proof. Let A denote the right hand side and B the left hand side of (5) and let θdenote the angle between the vectors a and b. By direct computation one verifiesthe identities

A = |a|p + |b|p − (|a|p−1 |b| + |b|p−1 |a|) cos θ

B = |a|p + |b|p − 2 |a|p2 |b|

p2 cos θ.

From the elementary inequality

tp−1 + t ≤ 2p tp2 + (1 − 1

p )(tp + 1

)we deduce that

|a|p−1 |b| + |b|p−1 |a| ≤ 2p |a|

p2 |b|

p2 + (1 − 1

p )(|a|p + |b|p

).

Since the case cos θ ≤ 0 is trivial we can assume cos θ ≥ 0. Using the above estimatewe obtain

A ≥ 1p

(|a|p + |b|p

)− 2

p |a|p2 |b|

p2 cos θ = 1

pB.

APPENDIX TO A 5

Lemma 2.3. Assume p ≥ 2. There exists a constant γ > 0 such that(|a|p−2

a − |b|p−2b)· (a − b) ≥ γ(|a| + |b|)p−2 |a − b|2

holds for all a, b ∈ RN .

Proof. Following the calculations in the proof of Lemma 2.1, we obtain(|a|p−2

a − |b|p−2b)· (a − b) ≥ |a − b|2

∫ 1

0

|sa + (1 − s)b|p−2ds

= (|a| + |b|)p−2 |a − b|2∫ 1

0

f(s)p−22 ds,

where

f(s) = s2α2 + (1 − s)2β2 + 2s(1 − s)αβ cos θ, (α, β) =(|a| , |b|)|a| + |b| .

Clearly α + β = 1, implying 0 ≤ f ≤ 1. Thus, assuming p ≥ 4,∫ 1

0

f(s)p−22 ds ≥

(∫ 1

0

f(s) ds

) p−22

=(

13α2 + 1

3β2 + 13αβ cos θ

) p−22

=(

13 + 1

3αβ(cos θ − 2)) p−2

2 ≥(

13 + 1

12 (cos θ − 2)) p−2

2

≥(

112

)p−22

Corollary 2.4. Assume p ≥ 2. There exists a constant C > 0 such that

|a − b|p ≤ C∣∣∣|a| p−2

2 a − |b|p−22 b∣∣∣2

for all a, b ∈ RN .

Proof. From Lemma 2.3, upon substituting p → p+22 and applying the Cauchy-

Schwarz inequality, we obtain

|a − b|p2 ≤ γ−1

∣∣∣|a| p−22 a − |b|

p−22 b∣∣∣ .

2.2. The case 1 < p < 2. Given f ∈ Lp(TN ) with zero mean value, we can find∇u such that∫

TN

|∇u|p−2 ∇u · ∇v dx =∫TN

fv dx ∀v ∈ W 1,q(TN ).

By choosing suitable test functions v, we obtain

(6)∫TN

Dh(|∇u|p−2 ∇u

)· Dh(∇u) dx = −

∫TN

fD−h(Dhu) dx,

Dhu denoting the difference quotient

Dhu(x) =u(x + hw) − u(x)

h,

where 0 < h < 1 and w is an arbitrary direction in SN−1. Applying Lemma 2.1yields

Dh(|∇u|p−2 ∇u

)· Dh(∇u) ≥ θEp−2

∣∣Dh(∇u)∣∣2 ,

6 J. FABRICIUS

where θ > 0 is a constant and

E(x) = |∇u(x)| + |∇u(x + hw)|Moreover, the elementary inequality tp ≤ t2 + 1

4 implies∣∣Dh(∇u)∣∣p ≤ |∇u|p−2 ∣∣Dh(∇u)

∣∣2 + 14 |∇u|p

≤ Ep−2∣∣Dh(∇u)

∣∣2 + 14 |∇u|p

Thus

(7)∫TN

Dh(|∇u|p−2 ∇u

)· Dh(∇u) dx ≥ θ

∫TN

∣∣Dh(∇u)∣∣p dx − θ

4

∫TN

|∇u|p dx.

We estimate the righ hand side of (6) with Young’s inequality∣∣∣∣∫TN

D−h(Dhu) dx

∣∣∣∣ ≤ ε1−qq

∫TN

|f |q dx + εp

∫TN

∣∣D−h(Dhu)∣∣p dx

≤ ε1−qq

∫TN

|f |q dx + εpC

∫TN

∣∣Dh(∇u)∣∣p dx,

(8)

for any ε > 0. Choosing ε = pθ2C , we combine (7) and (8) to obtain

θ2

∫TN

∣∣Dh(∇u)∣∣p dx ≤ θ

4

∫TN

|∇u|p dx + C

∫TN

|f |q dx.

This gives us enough information to conclude that uxi ∈ W 1,p(TN ), i = 1, . . . , N .Summing up, we have proved the following theorem.

Theorem 2.5. Assume 1 < p < 2. Then a weak solution u ∈ W 1,p(TN ) of thep-Poisson equation (2) actually belongs to W 2,p(TN ).

2.3. The case p ≥ 2. In view of Lemma 2.2 we can estimate∫TN

Dh(|∇u|p−2 ∇u

)· Dh(∇u) dx ≥ 1

p

∫TN

∣∣DhW∣∣2 dx.

where W = |∇u|p−22 ∇u.

From (6), (8) and Lemma 2.3 we deduce that there exists a constant C > 0 suchthat, ∫

TN

∣∣DhW∣∣2 dx ≤ (p − 1)ε1−q

∫TN

|f |q dx + Cε

∫TN

∣∣DhW∣∣2 dx.

Choosing ε = 12C we obtain∫

TN

∣∣DhW∣∣2 dx ≤ C ′

∫TN

|f |q dx.

Thus Wi ∈ W 1,2(TN ). Summing up, we have proved

Theorem 2.6. Assume p ≥ 2. Then a weak solution u ∈ W 1,p(TN ) of the p-Poisson equation (2) satisfies |∇u|

p−22 ∇u ∈ W 1,2(TN )N .

References

[1] P. Lindqvist. Notes on the p-Laplace equation. Report. Department of Mathematics and Sta-

tistics, 102, University of Jyvaskyla, 2006.

Paper B

REITERATED HOMOGENIZATION APPLIED INHYDRODYNAMIC LUBRICATION


Abstract. This work is devoted to studying the combined effect that arisesdue to surface texture and surface roughness in hydrodynamic lubrication. Aneffective approach to analyzing this problem is through theory of reiteratedhomogenization with three scales. In the numerical analysis of such prob-lems, a very fine mesh is needed, suggesting some type of averaging. To thisend a general class of problems is studied that, e.g., includes the incompress-ible Reynolds problem in both artesian and cylindrical coordinate forms. Todemonstrate the effectiveness of our method several numerical results are pre-sented that clearly show the convergence of the deterministic solutions towardthe homogenized solution. Moreover, the convergence of the friction force andthe load carrying capacity of the lubricant film is also addressed in this paper.In conclusion, reiterated homogenization is a feasible mathematical tool thatfacilitates the analysis of this type of problem.

1. Introduction

Throughout the years, the general theory of homogenization has been success-fully applied to different problems connected to hydrodynamic lubrication, see e.g.[1, 2, 3, 4, 5, 6, 7]. In these works it was shown that the rapid oscillations (in the co-efficients of the Reynolds type equation under consideration) induced by the surfaceroughness, could efficiently be averaged by the homogenization method employed.In these previous results, it is assumed that the lubrication problem exhibits twoseparable scales, i.e. a global scale describing the geometric shape of the applicationand a local scale describing the surface roughness.

In the present work it is assumed that the problem of interest, exhibits threeseparable scales, i.e. one global scale describing geometry, one oscillating localscale describing the surface texture and a faster oscillating local scale describingthe surface roughness. Homogenization of problems with two or more oscillatingscales are referred to as reiterated homogenization, see e.g. [8, 9, 10]. In this paper,a generalized form of the Reynolds problem is considered, governing incompress-ible and Newtonian flow, with the advantage to unify both the Cartesian and thecylindrical coordinate formulations. In particular, the aim is to obtain a generalhomogenized problem that corresponds to a class of problems modelled by (1). Onetechnique within the homogenization theory is the formal method of multiple scaleexpansion, see e.g. [8, 11]. To accomplish this aim the formal method of multi-ple scale expansion is employed to obtain a homogenized problem (8) for (1). For

Key words and phrases. Reynolds equation, reiterated homogenization, surface roughness andtexture.

1


other problems connected to the incompressible Reynolds that have been studiedby multiple scale expansion see [1, 3, 7].

By means of numerical analysis, the convergence, of the direct numerical solu-tion toward the homogenized counterpart, in terms of load carrying capacity andhydrodynamically induced friction is quantified. The results show that the com-bined effect due to texture and roughness on a modelled bearing can be effectivelyanalyzed through reiterated homogenization. More specifically, the discrepancies,between the proposed method and the direct numerical approach, in terms of pre-dicted load carrying capacity and friction force are tolerably small; O (1%) fortextures as well as roughness of wavelengths likely to be found in a real application.That is, wavelengths within the ranges 1/100 − 1/10 of the length bearing for thetexture and 1/10000 − 1/100 for the roughness.

2. The homogenization procedure

In this section a class of equations that includes the Reynolds equation governingincompressible Newtonian flow is considered. It can be seen that the generalizedform (1), makes it possible to study the Reynolds problem in its Cartesian, andcylindrical coordinate forms. (See Section 4)

Let Ω be an open bounded subset of R2, Y = (0, 1)2 and Z = (0, 1)2. Introducethe auxiliary matrix A =(aij), where aij = aij(x, y, z), and i = 1, 2, and j = 1, 2 aresmooth functions that are Y -periodic in y and Z-periodic in z. It is also assumedthat a constant α > 0 exists such that

2∑i,j=1

aij(x, y, z)ξiξj ≥ α |ξ|2 for every ξ ∈ R2.

Moreover, we introduce the auxiliary vector b = (bi), where bi = bi(x, y, z) andi = 1, 2, are smooth functions that are Y -periodic in y and Z-periodic in z. Letε > 0 and define the matrix Aε and the vector bε as

Aε(x) =(

aε11(x) aε

12(x)aε21(x) aε

22(x)

)= A(x, x/ε, x/ε2),

bε(x) =(

bε1(x)

bε2(x)

)= b(x, x/ε, x/ε2).

Consider the following boundary value problem

∇x · (Aε(x)∇xpε(x)) = ∇x · bε(x) in Ω,(1)

pε(x) = 0 on ∂Ω.

For small values of the parameter ε the coefficients in (1) are rapidly oscillating.This suggests some type of asymptotic analysis. We will see that pε → p0 as ε → 0and that p0 can be found by solving a so called homogenized equation (8), whichdoes not contain any rapid oscillations. This means that p0 may be used as anapproximation of the solution pε for small values of ε.

We will use the method of multiple scale expansion developed in the homoge-nization theory to derive a homogenization result connected to (1). For general in-formation concerning this method in connection to homogenization, see e.g. [8, 11].The homogenization of Reynolds type equations involving only one local scale havebeen studied by multiple scale expansion in [1, 3, 7, 12].

REITERATED HOMOGENIZATION APPLIED IN HYDRODYNAMIC LUBRICATION 3

Assume that pε is of the form

(2) pε(x) =∞∑

i=0

εipi(x, x/ε, x/ε2),

where pi = pi(x, y, z) is Y -periodic in y and Z-periodic in z. The main idea is toinsert the expansion (2) into (1), and then collect terms of the same order of ε andanalyze the system of equations obtained. A comprehensive analysis can be foundin Appendix B. The main result is that the leading term p0 in the expansion (2) isof the form p0 = p0(x) and is found by the following homogenization algorithm:

Step 1: Solve the local problems (on the z-scale)

(3) ∇z · (A (∇zui + ei)) = 0 in Z, (i = 1, 2) ,

(4) ∇z · (A∇zu0 − b) = 0 in Z.

Here ui = ui(x, y, z), i = 0, 1, 2, is Y -periodic in y, Z-periodic in z and e1, e2 isthe canonical basis in R2. Use these local solutions to define the matrix

A = A (x, y, z) =

⎛⎜⎝ 1 +∂u1

∂z1

∂u2

∂z1∂u1

∂z21 +

∂u2

∂z2

⎞⎟⎠ .

Step 2: Solve the local problems (on the y-scale)

(5) ∇y ·(AAz

(∇yvi + ei))

= 0 in Y, (i = 1, 2) ,

(6) ∇y ·(AAz∇yv0 −

(b − A∇zu0

z))

= 0 in Y.

Here vi = vi(x, y), i = 0, 1, 2, is Y -periodic in y and AAzis the average with respect

to Z. Use these local solutions to define the matrix

B = B (x, y) =

⎛⎜⎝ 1 +∂v1

∂y1

∂v2

∂y1∂v1

∂y21 +

∂v2

∂y2

⎞⎟⎠ .

Step 3: Compute the homogenized matrix A0 and the homogenized vector b0

by the following formulas

(7) A0(x) = AABzy

and b0(x) = b − A∇zu0 − AA∇yv0zy

.

Step 4: Find p0 by solving the so called homogenized problem

∇x · (A0(x)∇xp0(x)) = ∇x · b0(x) in Ω,(8)

p0(x) = 0 on ∂Ω.

The main advantage of the above algorithm is that the scales are treated sep-arately, i.e. first one ”averages” with respect to the z-scale, then with respect tothe y-scale and finally one solves the homogenized equation. We note that thehomogenized equation does not contain any oscillating coefficents, nevertheless, ittakes into account the effects of the local scales, see (7). The fact that the scalescan be separated in this way, significantly simplifies the numerical analysis of theproblem.


3. An additional result

In this section, the convergence of ∇pε is investigated. The functions pi, i =0, 1, 2, in the expansion is of the form,

p0 = p0(x), p1 = p1(x, y), p2 = p2(x, y, z),

see Appendix B. When inserted into (2) we see that

∇xpε (x) = ∇xp0 (x) + ∇yp1 (x, y) + ∇zp2 (x, y, z) + ε [. . .] ,

which means that

∇xpε (x) ≈ ∇xp0 (x) + ∇yp1 (x, y) + ∇zp2 (x, y, z)

for small values of ε. According to the analysis in Appendix B, p1 and p2 can beexpressed in terms of the solutions ui and vi of the local problems (3), (4), (5) and(6), respectively. Making use of (49) and (53) in addition to (45) and (58) yields,

(9) ∇xpε (x) ≈ ∇zu0 (x, y, z)+A (x, y, z)∇yv0 (x, y)+A (x, y, z)B (x, y)∇xp0 (x) ,

after some straightforward calculations. According to [10, 14, 15], the followingconvergence holds∫

Ω

∇xpε (x) ϕ(x, x/ε, x/ε2) dx −→(10) ∫Ω

∫Y

∫Z

[∇zu0 + A∇yv0 + AB∇xp0] ϕ (x, y, z) dzdydx,

for any smooth function ϕ that is Y -periodic in y and Z-periodic in z.

4. Application to hydrodynamic lubrication

In this section we study how our general reiterated homogenization result canbe applied to analyze the effects of texture and surface roughness in hydrodynamiclubrication goverened by the Reynolds equation. For this purpose we introduce anauxiliary function that may be used to represent the lubricant film thickness

(11) h(x, y, z) = h0(x) + hT (x, y) + hR(x, y, z),

where• h0(x) describes the geometry of the bearing,• hT (x, y) is a Y -periodic function in y, representing surface texture,• hR(x, y, z) is a Y -periodic function in y and a Z-periodic in z, representing

the roughness contribution.Note that this formulation admits studying problem where the texture and the

roughness changes with position at the tribological interface. For example, thisenables studying the effects of a texture only on a part of the surface, which in turnmay exhibit different surface roughness patterns at different parts of the textureitself. However, here the numerical examples are restricted to consider the casewhere the texture and the roughness representation does not change with the po-sition, i.e., hT = hT (y) and hR = hR (z). By making use of the auxiliary functionh, it is possible to model the deterministic film thickness hε as

(12) hε(x) = h(x, x/ε, x/ε2) = h0(x) + hT (x/ε) + hR(x/ε2),

where ε is a parameter that describes the texture and roughness wavelength.


Now, by choosing

Aε(x) =(

h3ε(x) 00 h3

ε(x)

),(13a)

bε(x) = 6μUhε (x) e1,(13b)

in (1), where e1 = (1, 0), we obtain the Reynolds equation describing incompressibleNewtonian flow in Cartesian coordinates, i.e.,

∇x ·((

h3ε 00 h3

ε

)∇xpε

)= 6μU∇x · (hεe1) in Ω,(14)

pε(x) = 0 on ∂Ω.

Here, pε is the hydrodynamically induced pressure distribution, μ is the (constant)viscosity of the Newtonian lubricant and U is the linear speed of the moving smoothsurface.

We also observe that by choosing

Aε(x) =(

h3ε(x)/x2 0

0 x2h3ε(x)

),(15a)

bε(x) = 6μωx2hε (x) e1,(15b)

in (1), where ω is the angular speed of the smooth rotating surface and (x1, x2) arethe angular and the radial coordinates we obtain the Reynolds equation describingincompressible Newtonian flow in cylindrical coordinates

∇x ·((

h3ε(x)/x2 0

0 x2h3ε(x)

)∇xpε

)= 6μωx2∇x · (hεe1) in Ω,(16)

pε(x) = 0 on ∂Ω.

It should be noted that our homogenization result, which is that pε −→ p0 asε −→ 0, do not require any restrictions on the geometry, neither on the texture(y-scale) nor on the roughness (z-scale). The only limitation is that ε should besufficiently small in order to approximate the hydrodynamic pressure pε with p0. Aswill be seen this is actually no limitation since ε is very small in realistic examples.

From the homogenization result, convergence of load carrying capacity Iε auto-matically follows, i.e.

Iε =∫

Ω

pε (x) dx −→∫

Ω

p0 (x) dx = I0.

Moreover, we studied the convergence of ∇pε in Section 3. The convergence of hy-drodynamically induced friction force, Fε, and frictional torque, Tε, are connectedto the derivative ∂pε/∂x1, and by making use of (10) we obtain the following ex-pressions

(17) Fε =∫

Ω

(μU

hε (x)+

hε(x)2

∂pε

∂x1

)dx −→

F0 =∫

Ω

∫Y

∫Z

(μU

h (x, y, z)+

h (x, y, z)2

∂p0

∂x1

)dzdydx

+∫

Ω

∫Y

∫Z

h (x, y, z)2

[∂u0

∂z1+

∂u1

∂z1

∂p0

∂x1+

∂u2

∂z2

∂p0

∂x2

]dzdydx

+∫

Ω

∫Y

∫Z

h (x, y, z)2

[∂v0

∂y1+

∂v1

∂y1

∂p0

∂x1+

∂v2

∂y2

∂p0

∂x2

]dzdydx


+∫

Ω

∫Y

∫Z

h (x, y, z)2

⎡⎢⎣⎛⎜⎝

∂u1

∂z1∂u2

∂z1

⎞⎟⎠ ·

⎛⎜⎝⎛⎜⎝

∂v0

∂y1∂v0

∂y2

⎞⎟⎠+

⎛⎜⎝∂v1

∂y1

∂v2

∂y1∂v1

∂y2

∂v2

∂y2

⎞⎟⎠⎛⎜⎝

∂p0

∂x1∂p0

∂x2

⎞⎟⎠⎞⎟⎠⎤⎥⎦ dzdydx

for friction force and

(18) Tε =∫

Ω

x2

(μωx2

hε+

hε

2x2

∂pε

∂x1

)x2 dx1dx2 −→

T0 =∫

Ω

∫Y

∫Z

x2

(μωx2

h+

h

2x2

∂p0

∂x1

)dzdy

x2 dx1dx2

+∫

Ω

∫Y

∫Z

h

2

[∂u0

∂z1+

∂u1

∂z1

∂p0

∂x1+

∂u2

∂z2

∂p0

∂x2

]dzdy

x2 dx1dx2

+∫

Ω

∫Y

∫Z

h

2

[∂v0

∂y1+

∂v1

∂y1

∂p0

∂x1+

∂v2

∂y2

∂p0

∂x2

]dzdy

x2 dx1dx2

+∫

Ω

⎧⎪⎨⎪⎩∫

Y

∫Z

h

2

⎛⎜⎝∂u1

∂z1∂u2

∂z1

⎞⎟⎠ ·

⎛⎜⎝⎛⎜⎝

∂v0

∂y1∂v0

∂y2

⎞⎟⎠+

⎛⎜⎝∂v1

∂y1

∂v2

∂y1∂v1

∂y2

∂v2

∂y2

⎞⎟⎠⎛⎜⎝

∂p0

∂x1∂p0

∂x2

⎞⎟⎠⎞⎟⎠ dzdy

⎫⎪⎬⎪⎭x2 dx1dx2

for frictional torque. To clarify, from the equations above, the resulting homoge-nized quantity is made up of friction force/torque due to the smooth (averaged)film thickness plus a corrector term identified by three separate contributions, i.e.due to roughness or texture acting alone or roughness and texture acting together.

In the following, we conduct numerical investigations into the convergence asso-ciated with load carrying capacity and the hydrodynamically induced friction forceby employing a second order finite difference scheme. The results of these inves-tigations, justify the applicability of the homogenization process presented in thispaper. Subsequently, we study the effects of periodic texture and surface roughnessby considering a thrust pad bearing problem. It is observed that for one-dimensionaltexture and roughness representation only very small differences exist between thehomogenized numerical solution (HNS) and the direct numerical solution (DNS).We point out that it is only possible to find the DNS in the case of transversaland longitudinal (i.e. one-dimensional) texture and roughness due to the enormousnumber of discretization points that is required in the general case. From the gen-eral analysis, it is clear that it is always possible to obtain an approximate solutionp0 of pε, with very high accuracy by solving the homogenized equation. From anapplication point of view this means that for arbitrary (i.e. also two-dimensional)yet physically relevant, texture and roughness, a highly accurate approximation p0

of the pressure solution pε, can be obtained by solving the homogenized equation.This is one of the benefits with our method.

4.1. A numerical investigation of convergence. Computationally, it is ex-tremely demanding to retrieve the DNS for short wavelength roughness (and tex-ture). Therefore, to assess and quantify convergence, the one-dimensional problemwas first revisited. This elementary problem constitutes an excellent benchmarkfor the implemented numerics, since it is possible to obtain closed form expres-sions for the coefficients in the homogenized equation. Specifically, we obtain a


one-dimensional representation of the Reynolds equation, for incompressible andNewtonian flow, in Cartesian coordinates by considering (14), i.e.

d

dx

(h3

ε(x)dpε

dx(x))

= 6μUdhε

dx(x) in 0 ≤ x ≤ L,(19)

pε(0) = pε(L) = 0.

Here, L is the length of the stationary surface exhibiting texture and roughness.Through (11), the film thickness function of the modeled linear slider bearing,

is described with

h0 (x) = hmin +hmin

4

(1 − x

L

),

hT (s) = 2hR (s) =hmin

4

(12

(1 − cos (2πs)))

,

where hmin denotes the fixed minimum film thickness of the corresponding smoothproblem, i.e. the problem with a smooth stationary surface as well as a smoothmoving surface. To generalize the results, the dimensionless variables X = x/L,H = h/hmin and Pε = pε/

(6μUL/h2

min

)were introduced to obtain a dimensionless

Reynolds problem,

d

dX

(H3

ε (X)dPε

dX(X))

=dHε

dX(X) , in 0 ≤ X ≤ 1,(20)

Pε(0) = Pε(1) = 0.

We also present the dimensionless representation of the auxiliary film thicknessfunction, in terms of these dimensionless variables, i.e.

H (X, y, z) = 1 +14

(1 − X) +14

(12

(1 − cos (2πy)))

+18

(12

(1 − cos (2πz)))

.

The homogenized problem corresponding to (20) reads as

d

dX

(1

H−3 (X, y, z)zy

dP0

dX

)=

d

dX

⎛⎝H−2 (X, y, z)zy

H−3 (X, y, z)zy

⎞⎠ , in 0 ≤ X ≤ 1,(21)

P0(0) = P0(1) = 0.

Figure 1 illustrates the convergence of load carrying capacity Iε toward I0 withdecreasing ε. In fact, it is the measure

(22) |Iε − I0|/ I0,

(equivalent to ∫ 1

0

|Pε (X) − P0 (X)| dX

/∫ 1

0

P0 (X) dX,

if Pε, P0 ≥ 0) that is considered as being a function of ε in the figure. Whencomputing the DNS, 25 discrete nodes were used to represent a single wavelengthof the texture, e.g., for ε = 2−7, a total number of

(1/2−7

)2 25 = 219 grid nodes wereused. As deduced from the figure, the rate of convergence is very close to linear,with the goodness of fit equaling 0.99. To further elaborate on the convergence ofPε towards P0, we illustrate a set of DNS (Pε) and the HNS (P0) in Figure 2.

To facilitate the derivation of the specific version of (17) that corresponds to theone-dimensional dimensionless form of (1), we first choose Aε = H3

ε (x1) and bε =


7 6 5 4 30

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

log2ε

|Iε−I

0|/I

0

Figure 1. Convergence of load carrying capacity Iε towards I0

with decreasing ε.

0 0.2 0.4 0.6 0.8 10

0.004

0.008

0.012

0.016

P

X

Pε=2

3

Pε=2

5

Pε=2

7

P0

Figure 2. A set of DNS (Pε) and the HNS (P0).

Hε(x1). Then, owing to (17), we have the following convergence, in terms of di-mensionless friction force Fε = Fε /(μUL/hmin) ,

(23) Fε =∫ 1

0

1Hε (X)

+ 3Hε (X)dPε

dX(X) dX −→


F0 =∫ 1

0

∫ 1

0

∫ 1

0

(1H

+ 3Hdp0

dX

)dzdydX

+∫ 1

0

∫ 1

0

∫ 1

0

3H

(∂u0

∂z+

∂u1

∂z

dp0

dX

)dzdydX

+∫ 1

0

∫ 1

0

∫ 1

0

3H

(∂v0

∂y+

∂v1

∂y

dp0

dX

)dzdydX

(24) +∫ 1

0

∫ 1

0

∫ 1

0

3H

(∂u1

∂z

(∂v0

∂y+

∂v1

∂y

dp0

dX

))dzdydX

For the one-dimensional problem we can solve the cell problems (3), (4), (5) and(6) explicitly. Inserting the solutions

∂u0

∂z= H−2 (X, y, z) − H−2(X,y,z)

z

H−3(X,y,z)zH−3 (X, y, z)

∂v0

∂y= H−2 (X, y, z)

z − H−2(X,y,z)zy

H−3(X,y,z)zyH−3 (X, y, z)

z

∂u1

∂z= −1 + H−3(X,y,z)

H−3(X,y,z)z

∂v1

∂y= −1 + H−3(X,y,z)

z

H−3(X,y,z)zy

into F0 we find that

F0 =∫ 1

0

∫ 1

0

∫ 1

0

1H (X, y, z)

+ 3H (X, y, z)

×[H−2 (X, y, z) − H−2 (X, y, z)

z

H−3 (X, y, z)z H−3 (X, y, z)

+H−3 (X, y, z)

H−3 (X, y, z)z

⎛⎝H−2 (X, y, z)z −⎛⎝H−2 (X, y, z)

zy

H−3 (X, y, z)zy

⎞⎠H−3 (X, y, z)z

⎞⎠+

H−3 (X, y, z)

H−3 (X, y, z)z

H−3 (X, y, z)z

H−3 (X, y, z)zy

dP0

dX

]dzdydX.

Figure 3 displays the convergence of Fε. Actually, Figure 3 visualizes the variationwith ε in the expression

(25) |Fε − F0|/ F0 = |Fε −F0|/F0.

In comparison to the (almost) linear convergence for Iε, the rate of convergenceof Fε is lower than linear, according to the figure. However, the results presentedabove, particularly those shown in Figures 1 and 3, clearly serve as justificationof the applicability of the proposed reiterated homogenization result. More specif-ically, the discrepancies in terms of predicted load carrying capacity and frictionforce are tolerably small; O (1%) for textures as well as roughness of wavelengthslikely to be found in a real application. That is, wavelengths within the ranges1/100 − 1/10 of the length bearing for the texture and 1/10000 − 1/100 for theroughness.


7 6 5 4 35.6

5.8

6

6.2

6.4

6.6

6.8 x 10 4

log2ε

|Fε−F

0|/

F0

Figure 3. Convergence of friction force Fε toward F0 with de-creasing ε

4.2. Application to a thrust pad bearing problem. The effects of periodictexture and surface roughness are here exemplified by considering a thrust padbearing problem. The flow is assumed to be modelled through the cylindricalcoordinate formulation of the Reynolds problem, i.e. (16). A point x in the bearingis identified by its cylindrical coordinates x = (x1, x2) ∈ Ω = [−θ0/2, θ0/2] ×[R, 2R] (with x1 denoting the angular and x2 the radial coordinate). In this caseconvergence of frictional torque, Tε, is given by (18).

There are two ways of approaching the lubrication problem. In the precedingsection, we regarded the separation hmin between the surfaces on the global scaleas an input parameter and retrieved the solution in terms of the single dependentparameter, i.e. dimensionless hydrodynamic pressure Pε, by solving the Reynoldsequation (20). Observe that due to the specific dimensionless formulation chosenwe obtained the solution pε for arbitrary hmin > 0.

In approaching the present thrust pad bearing problem, we employ a force-balance equation

(26) W −∫

Ω

pε (x) x2dx2dx1 = 0,

where the applied load W appears as an input parameter. We then solve theReynolds equation (16) and the force-balance equation (26) to retrieve the solutionin terms of the two dependent parameters, namely the separation between thesurfaces on the global scale h00 and the hydrodynamic pressure pε. Again, we use(11) to represent the film thickness and define

(27) h0 (x) = h00 −x2 (sin(x1)− sin (θ0))

R sin(θ0)θ0Rtan α,


h0(x)

x1, ω

x2

Figure 4. A schematic descriptions of a single pad.

to model a single bearing segment. Note that h00 exactly defines the height of theparallel gap between the trailing edge and the rotating shaft surface and that itsvalue depends on the applied load W. (For the smooth problem (hT = hR ≡ 0),h00 represents the minimum film thickness.) In (27), R denotes the inner radius,θ0 defines the size of the pad in radians and α controls the pad inclination. SeeFigure 4 for a schematic description of a bearing segment within the bearing.

By considering the homogenized correspondence (8) to (16), we can proceedto examine the effects of periodic texture and surface roughness . The case oftransversal sinusoidal surface texture as well as surface roughness is addressed first,

hε (x) = hε00 −

x2 (sin(x1)− sin (θ0))R sin(θ0)

θ0Rtan α+

hεT (x) + hε

R (x) ,(28)

wherehε

T (x) := hT (x/ε) and hεR (x) := hR

(x/ε2

).

Explicitly, the auxiliary functions are

(29) hT (y) =aT

2(1− cos (2πy1))

and

(30) hR (z) =aR

2(1− cos (2πz1)) .

The separation hε00 is regarded as a parameter that is parameterized in ε and

dependent on W, and we can therefore solve the Reynolds equation (16) and theforce-balance criterion (26) for hε

00 and pε (or h000 and p0 for the corresponding

homogenized system of equations). The input parameters chosen for this specificproblem are found in Table 1.

To resolve the direct numerical solution (DNS) properly, each roughness wave-length is resolved with 25 discrete nodes. For the results presented here, this meansa total number of uniformly distributed nodes of 25

(24)2 = 213 in the x1-direction

for ε = 2−4, while 26 nodes were considered sufficient for the discretization in thex2-direction. The coefficients in the homogenized matrix and vector, both given in


Table 1. Input parameters

Parameter Description Value Unitθ0 Pad size (in degrees) 25 degR Pad inner radius 10 · 10−3 mμ Fluid viscosity 0.3 Pa sω Smooth surf. angular speed 2.5 rad/sα Pad inclination 1.6 · 10−4 radW Applied load 10 Na Roughness amp. scaling param. 0.5 · 10−6 m

Table 2. Normalized homogenized property h000/hs

00, transversalsinusoidal texture and roughness.

aT \ aR 0 a 2a 4a 8a0 1.0000 0.9639 0.9298 0.8677 0.7636a 0.9639 0.9278 0.8937 0.8317 0.72772a 0.9298 0.8937 0.8597 0.7978 0.69424a 0.8677 0.8317 0.7979 0.7363 0.63418a 0.7636 0.7277 0.6942 0.6341 0.5364

(7), were obtained by solving the (one-dimensional) periodic Y or Z cell problemswith 26 nodes in the y1- and the z1- directions.

Table 2 displays normalized homogenized separation h000/hs

00, where hs00 = 6.72 ·

10−6m denotes the minimum film thickness for the correspondingly smooth prob-lem. In the table, texture amplitude aT increases vertically downwards, whileroughness amplitude aR increases horizontally to the right, as indicated by aT \ aR.

For ε = 2−4 the maximum relative difference between hε00 and h0

00 was foundto be less than 0.01. More specifically, for ε = 2−4, corresponding to a texturewavelength θ02R/24 ≈ 0.5 · 10−3m measured at the outer radius (x2 = 2R) andfor (aT , aR) = (8a, 8a), we obtained

∣∣hε00 − h0

00

∣∣/h000 = 0.0097. The fact that the

maximum difference occurs for (aT , aR) = (8a, 8a) is to be expected, as an increasein texture amplitude or roughness amplitude also increases the discretization errors.Since, in theory, it makes sense to distinguish between roughness and texture onlywhen both of them are present, the first row and column in Table 2 could beused as a benchmark of the numerical routine employed. Although the figuresin the table seem to indicate that the rigid body separation is symmetrical withrespect to texture and roughness amplitude, no theoretical evidence supporting thisis reported here.

Table 3 presents the variation in the normalized homogenized frictional torque,T0/Ts. The numerical values of T0 are computed from (18) and the frictional torqueexhibited for a set of perfectly smooth surfaces, is found to be Ts = 1.16 · 10−3Nm.Also, for ε = 2−4 and (aT , aR) = (8a, 8a) we find that |Tε − T0|/ T0 = 0.0037.

According to Table 3, the previously remarked symmetry observed in Table 2,with respect to texture and roughness amplitude also to hold true for the homog-enized frictional torque. For example, a texture of amplitude 2a combined withroughness of amplitude 0, i.e. (aT , aR) = (2a, 0) , and a texture of amplitude 0 com-bined with roughness of amplitude 2a, i.e. (aT , aR) = (0, 2a) , yields approximately


Table 3. Normalized homogenized frictional torque T0/Ts,transversal sinusoidal texture and roughness.

aT \ aR 0 a 2a 4a 8a0 1.0000 1.0005 1.0021 1.0083 1.0312a 1.0005 1.0011 1.0026 1.0087 1.03142a 1.0021 1.0027 1.0043 1.0106 1.03334a 1.0083 1.0089 1.0106 1.0170 1.04048a 1.0312 1.0318 1.0336 1.0403 1.0641


00, transversalsinusoidal texture and longitudinal sinusoidal roughness.

aT \ aR 0 a 2a 4a 8a0 1.0000 0.9627 0.9251 0.8493 0.6947a 0.9639 0.9266 0.8890 0.8132 0.65852a 0.9298 0.8925 0.8549 0.7790 0.62424a 0.8677 0.8304 0.7928 0.7167 0.56108a 0.7636 0.7262 0.6884 0.6115 0.4532

Table 5. Normalized homogenized frictional torque T0/Ts,transversal sinusoidal texture and longitudinal sinusoidal rough-ness.

aT \ aR 0 a 2a 4a 8a0 1.0000 1.0003 1.0013 1.0053 1.0219a 1.0005 1.0009 1.0019 1.0059 1.02252a 1.0021 1.0025 1.0035 1.0075 1.02424a 1.0083 1.0087 1.0098 1.0138 1.03058a 1.0312 1.0316 1.0327 1.0369 1.0541

the same h000 or T0 according to the tables, i.e. h0

00 = 0.9298 and T0 = 1.0021,whereas (aT , aR) = (a, a) results in h0

00 = 0.9278 and T0 = 1.0011. However, super-positioning the effects resulting from (aT , aR) = (a, 0) and (aT , aR) = (0, a) gives,with 4 decimal places, h0

00 = 0.927 8 and T0 = 1. 0010. The relative discrepanciesbetween the superpositioned results and the directly computed results were foundto be 3.15 · 10−6 for h0

00 and 1.35 · 10−5 for T0. For the frictional torque, we suggestthat this relative difference is attributed to the last term in (18), i.e. the term forthe combined effect of texture and roughness.

Next, we consider the textured pad from the preceding case, i.e. (28) and (29),but with a longitudinal instead of a transversal sinusoidally shaped surface rough-ness,

(31) hR (z) =aR

2(1− cos (2πz2)) .

The results are compiled in Tables 4 and 5.These tables illustrate how a longitudinally shaped roughness (or texture, inter-

preting the data in the first row as being induced by a surface texture instead ofsurface roughness) influences film formation to a higher degree than the transversal



00, differenttextures and roughnesses.

TType\RType Smooth Eq.(30) Eq.(31) Fig. 7 Fig. 8Smooth 1.0000 0.8054 0.7615 0.5641 0.7781Eq.(30) 0.7636 0.5752 0.5218 0.3355 0.5464Eq.(31) 0.6947 0.5002 0.4573 0.2596 0.4723Eq.(32) 0.9346 0.7420 0.6951 0.5012 0.7142Eq.(33) 0.9808 0.7866 0.7422 0.5454 0.7592

Table 7. Normalized homogenized property T0/Ts, different tex-tures and roughnesses.

TType\RType Smooth Eq.(30) Eq.(31) Fig. 7 Fig. 8Smooth 1.0000 1.0199 1.0133 1.0245 1.0197Eq.(30) 1.0312 1.0524 1.0451 1.0573 1.0521Eq.(31) 1.0219 1.0471 1.0370 1.0534 1.0458Eq.(32) 1.0143 1.0349 1.0281 1.0396 1.0347Eq.(33) 1.0027 1.0228 1.0161 1.0274 1.0226

correspondence. When considering the induced frictional torque, the effects causedby the longitudinally shaped roughness (or texture) shows a less pronounced effectthan that of the corresponding transversal case. This corresponds well with whatwould be intuitively expected and confirms what is already well-known within thefield.

Tables 6 and 7 compare the two more realistic surface roughness representationsfound in Figures 7 and 8 with the previously considered sinusoidal representationsas well as the smooth case. In addition to the transversal and longitudinal sinu-soidal textures, the textures given by (32) (displayed in Figure 5) and (33) (dis-played in Figure 6) were also considered. All four roughness representations werescaled to exhibit an average roughness value Ra

(=∫

Z

∣∣hR (z)−∫

ZhR (z) dz

∣∣ dz)

of 1μm. This means that the corresponding amplitude of the sinusoidal represen-tations (both the transversal and the longitudinal) become aR = Rz/2 = π μm,i.e., Rz = 2π μm ≈ 6.28μm. The rough surface in the Figure 7 has Rz = 6.10μmand the one in Figure 8 has Rz = 11.00μm. In all simulations (except for the casewithout any texture) the texture amplitude was held fixed, i.e. aT = 4μm.

Figure 5 presents the mathematical description of the surface texture given by

(32) hT (y) = 10−50(y1−1/2)2 cos (2π (y1 − 1/2)) ,

while the surface representation presented in Figure 6 is modelled mathematicallyby

(33) hT (y) = 10−25((y1−1/2)2+(y2−1/2)2) cos (2π (y1 − 1/2)) cos (2π (y2 − 1/2)) .

Figure 7 displays a surface roughness representation hR (z), exhibiting an almostunskewed striated pattern, while Figure 8 displays a negatively skewed surfaceroughness representation hR (z) that exhibits a reasonably random pattern. Bothof these roughnesses originate from measurements but have been re-sampled and


0

0.5

1

0

0.5

1

y1

y2

hT(y)

Figure 5. Artificially ground surface texture hT (y).

0

0.5

1

0

0.5

1

y1

y2

hT(y)

Figure 6. Artificially dimpled surface texture hT (y).

0

0.5

1

0

0.5

1

z1

z2

hR(z)

Figure 7. A surface roughness hR (z), exhibiting a striated pat-tern, originating from a surface measurement.


0

0.5

1

0

0.5

1

z1

z2

hR(z)

Figure 8. A negatively skewed surface roughness hR (z) exhibit-ing a reasonably random pattern, originating from a surface mea-surement.

normalized for the assessments conducted here. Normalized to an average rough-ness value, Ra = 1μm, these roughness representations have Rz = 6.10μm andRz = 11.00μm (as previously mentioned) and their corresponding skewness values,RSK = −0.0061 and RSK = −1.7284. In studying Table 6 one notices that thelongitudinal texture deteriorates film formation most, i.e. produces the smallestvalues of the ratio h0

00/hs00, and the artificially dimpled texture (33) the least with-

out considering the perfectly smooth surface. The surface roughness representationshown in Figure 7 is by far the most detrimental in terms of film formation. Thissurface roughness representation exhibits exactly the same Ra (ensured by the scal-ing) and approximately the same Rz and RSK values as those corresponding to thetransversal sinusoidal representation. The same table clearly shows that after theperfectly smooth surfaces, it is the transversal sinusoidal roughness representationthat generates the thickest film. Hence we conclude that for a prediction to bereliable it must consider more information than the three abovementioned surfaceroughness parameters.

Addressing frictional torque, it is - according to Table 7 - the artificially dim-pled texture (33) is again the texture inducing the smallest effect. However, it isthe transversal and not the longitudinal sinusoidal texture that influences frictionaltorque the most. In optimizing the performance in terms of film formation andinduced frictional torque, it is clear that perfectly smooth surfaces are preferred,this was also previously confirmed, see e.g. [3]. However, disregarding the unre-alistic perfectly smooth bearing, it is the artificially dimpled surface texture (33)that yields the thickest films and induces the smallest frictional torque. As well, itis the grounded surface roughness representation displayed in Figure 7 that clearlyhas the most severe influence on film formation and frictional torque. Thus, from amanufacturing point-of-view, in choosing from the selection of textures and rough-nesses found in Tables 6 and 7, it would probably be most convenient to use alaser dimpling technique to achieve the 4μm deep texture and then radially grindto a 1μm Ra-value. This would be a rather successful combination according tothe present findings. However, if the surface is further processed from its groundedstate, e.g. also chemically de-burred, it might display a surface finish similar to


that presented in Figure 8. In turn, this should facilitate film formation as well aslower the induced frictional torque, according to the results presented here.

5. Conclusions

Our main result is that we have successfully developed a reiterated homoge-nization procedure for a class of problems by using multiple scale expansion. Inparticular, the Reynolds problem, which governs incompressible and Newtonianflow in Cartesian and cylindrical coordinates, belongs to this class. This madeit possible to efficiently study problems connected to hydrodynamic lubrication in-cluding shape, texture and roughness. Herein lies the novelty of our results, whereasonly two scales, i.e. shape and roughness, have been considered previously we canconsider a third scale, i.e., the texture.

In addition, we have analyzed the convergence of the pressure gradient. Thisenabled us to study the limiting behavior of hydrodynamically induced frictionforce and frictional torque, as the wavelengths of the local scales tend to zero.

To demonstrate the applicability and effectiveness of our method, several numer-ical results are presented, which clearly show the convergence of the deterministicsolutions toward the homogenized solution. The quantification of convergence wasgiven in terms of load carrying capacity and friction force. In these convergenceillustrations only transversal and longitudinal roughness and texture were consid-ered. The reason for this was that it is impossible to obtain the full numericalsolution for two dimensional roughness and texture, due to enormous amount ofdiscretization points which are required to resolve the surface. However, by usingour homogenization result it is possible to study the effects of arbitrary roughnesswith very high accuracy by solving the derived smooth homogenized equation. Thiswas demonstrated in an example connected to a realistic thrust pad bearing prob-lem, where the effects of texture and roughness on film formation and frictionaltorque were investigated.

Based on the general convergence result for the pressure gradient, we were able todeduce the limit of the deterministic expression for the friction force. The resultinghomogenized quantity is made up of friction force due to the smooth (averaged)film thickness plus a corrector term. Moreover, in this corrector term, one canidentify three separate contributions, i.e. due to either roughness or texture actingalone or texture and roughness acting together. The presence of terms of the latterkind implies that roughness could enhance (or diminish) certain effects that areessentially due to texture (and vice versa). Our numerical results indicate thatthe combined effect due to texture and roughness on the modelled hydrodynamicbearings can be efficiently analyzed using reiterated homogenization. The resultingdiscrepancies in terms of predicted load carrying capacity and friction force aresmall; O (1%) for textures as well as roughnesses of wavelengths likely to be foundin a real application. That is, wavelengths within the ranges 1/100 − 1/10 of thelength bearing for the texture and 1/10000− 1/100 for roughness.

From the assessment of the combined effects of texture and roughness - that arisein the modelled thrust pad bearing - we adhere to the conclusion that reiteratedhomogenization is a feasible tool, and for any prediction to be reliable it mustconsider more information regarding the surface than the three surface roughnessparameters, Ra, Rz and RSK.


acknowledgment

This work was partly financed by:• The European Commission, Marie Curie Transfer of Knowledge Scheme

Predicting Lubricant Performance for Improved Efficiency (FP6).• The Swedish Research Council, An interdisciplinary study of rough surface

effects in lubrication by homogenization techniques, 621-2005-3168• Ghana Government Scholarship Secretariat.

References

[1] G. Bayada, J. B. Faure. A double scale analysis approach of the Reynolds roughness commentsand application to the journal bearing. J. Tribol. 111:3 (1989), 323-330.

[2] G. Bayada, S. Martin and C. Vazquez. An average flow model of the Reynolds roughnessincluding a mass-flow preserving cavitation model.J. Tribol. 127:4 (2005), 793-802.

[3] G. C. Buscaglia, I. Ciuperca, and M. Jai. The effect of periodic textures on the static char-acteristics of thrust bearings. J. Tribol. 127:4 (2005), 899-902.

[4] A. Almqvist, and J. Dasht. The homogenization process of the Reynolds equation describingcompressible liquid flow. J. Tribol. 39 (2006), 994-1002.

[5] M. Kane and B. Bou-Said. Comparison of homogenization and direct techniques for thetreatment of roughness in incompressible lubrication. J. Tribol. 126 (2004), 733-737.

[6] M. Kane and B. Bou-Said. A study of roughness and non-Newtonian effects in lubricatedcontacts. J. Tribol. 127 (2005), 575-581.

[7] A. Almqvist, E. K. Essel, L.-E. Persson, and P. Wall. Homogenization of the unstationaryincompressible Reynolds equation. Tribol. Int. 40(2007), 1344-1350.

[8] A. Bensousan, J.-L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structures.North-Holland, Amsterdam-New-York-Oxford, 1978.

[9] J.-L. Lions, D. Lukkassen, L.-E, Persson and P. Wall. Reiterated homogenization of nonlinearmonotone operators. Chin. Ann. Math. Ser. B. 22 (2001),1-12.

[10] G. Allaire, and M. Briane. Multiscale convergence and reiterated homogenization. Proc. R.Soc. Edinb. 126 (1996), 297-342.

[11] L.-E. Persson, L. Persson, N. Svanstedt, and J. Wyller. The homogenization method: Anintroduction. Lund: Studentlitteratur; 1993.

[12] A. Almqvist, R. Larsson, and P. Wall. The homogenization process of the time dependentReynolds equation describing compressible liquid flow. Tribologia – The Finnish Journal ofTribology, 26(2007), 30-44.

[13] D. Cioranescu, and P. Donato. An introduction to Homogenization. Oxford Lecture Series inMathematics and its Applications. Oxford Univ. Press, New York, 1999.

[14] J. L. Lions, D. Lukkassen, L. E. Persson, and P. Wall. Reiterated homogenization of nonlinearmonotone operators. Chin. Ann. of Math. 22B: 1 (2001), 1-12.

[15] D. Lukkassen, G. Nguetseng and P. Wall. Two scale convergence. Int. IJPAM . 2:1 (2002),35-86.


Appendix A. Notation

aij Elements of matrix AAε Deterministic matrixA0 Homogenized matrixAi Differential operator, i = 0, ..., 4bε Deterministic vectorb0 Homogenized vectorei Canonical basis in R2, e1 = (1, 0) and e2 = (0, 1)f

zAverage of f with respect to Z

(=∫

Zfdz)

fzy

Average of f with respect to Z and Y(=∫

Y

∫Z

fdzdy)

Fε Deterministic frictional forceF0 Homogenized frictional forceFε Dimensionless friction force = Fε/ (μUL/hmin)F0 Dimensionless homogenized friction forceh Auxiliary function used to modell film thicknessh0 Function describing global geometry of bearinghR Function describing the roughness part of film thicknesshε Deterministic film thicknesshT Function describing the texture part of film thicknesshmin Fixed minimum film thickness = min of h0

H Dimensionless film thickness = h/hmin

Iε Deterministic load carrying capacityI0 Homogenized load carrying capacityL Length of stationary surface exhibiting texture and roughnesspε Deterministic pressure solutionpi The ith term in the expansion of the pressure pε

p0 Homogenized pressure solutionPε Dimensionless deterministic pressure = pε

/(6μUL/h2

min

)Tε Deterministic frictional torqueT0 Homogenized frictional torqueTs Frictional torque for a perfectly smooth surface (= 1.16 · 10−3Nm)ui Z-periodic solution of the local problems, i = 0, 1, 2U Linear speed of moving surfacevi Y -periodic solution of the local problems, i = 0, 1, 2x Local spatial coordinate, x = (x1, x2)X Dimensionless spacial coordinate = x/Ly Local spatial coordinate, y = (y1, y2) = (x1/ε, x2/ε)Y Y -cell = [0, 1]2

z Local spatial coordinate, z = (z1, z2) =(x1/ε2, x2/ε2

)Z Z-cell = [0, 1]2

∂Ω Boundary of Ωε Parameter describing the roughness and texture scale (ε > 0)∇x Gradient operator, ∇x = ∇∇y Gradient operator, ∇y = (∂/∂y1, ∂/∂y2)∇z Gradient operator, ∇z = (∂/∂z1, ∂/∂z2)Ω Open bounded subset of R2


Appendix B

In this appendix the analysis leading to our homogenization result is presented,by deriving the homogenized equation (8) corresponding to (1). The method weuse is known as multiple scale expansion. For more information concerning thismethod in connection with homogenization see e.g. [8].

Let us first observe that the chain rule applied to a smooth function of the formψε(x) = ψ(x, y, z), where y = x/ε and z = x/ε2 gives that

(34) ∇xψε(x) =(∇x +

1ε∇y +

1ε2∇z

)ψ(x, y, z).

Inserting the expansion (2) (of pε) into (1) and making use of (34) we obtain(∇x +

1ε∇y +

1ε2∇z

)·[A(∇x +

1ε∇y +

1ε2∇z

) ∞∑i=0

εipi

]

=(∇x +

1ε∇y +

1ε2∇z

)· b.(35)

Let the differential operators Ai, i = 0, ..., 4 be defined as

A0 = ∇z · (A∇z) ,

A1 = ∇z · (A∇y) +∇y · (A∇z) ,

A2 = ∇x · (A∇z) +∇y · (A∇y) +∇z · (A∇x) ,

A3 = ∇x · (A∇y) +∇y · (A∇x) ,

A4 = ∇x · (A∇x) .

Using the above notation (35) may be written as(ε−4A0 + ε−3A1 + ε−2A2 + ε−1A3 + A4

) (p0 + εp1 + ε2p2 + ...

)=(ε−2∇z + ε−1∇y +∇x

)· b.

By comparing terms with the same order of ε (from−4 to 0), we obtain the followingsystem of equations

A0p0 = 0,(36a)

A0p1 + A1p0 = 0,(36b)

A0p2 + A1p1 + A2p0 = ∇z · b,(36c)

A0p3 + A1p2 + A2p1 + A3p0 = ∇y · b,(36d)

A0p4 + A1p3 + A2p2 + A3p1 + A4p0 = ∇x · b.(36e)

In the following, we make frequent use of the following well-known result:

Γu = f has a solution if and only if∫

Z

fdz = 0.(37)

In this case u is unique up to an additive constant.

Here, Γ is any of the operators A0, A1, A2, . . . and Z may be replaced with Y .See for example [11, p. 39] for a proof of (37). According to (37), it is clear thatp0 in (36a) does not depend on z, i.e.

(38) p0 = p0(x, y),


and this simplifies (36b) to

(39) A0p1(x, y, z) = −∇z · (A (x, y, z)∇yp0(x, y)) .

By linearity

(40) p1(x, y, z) = u1(x, y, z)∂p0

∂y1(x, y) + u2(x, y, z)

∂p0

∂y2(x, y) + p1(x, y),

where the Z-periodic function ui = ui(x, y, z), i = 1, 2 is a solution (unique up toa constant) to the following local problem

(41) ∇z · (A (∇zui + ei)) = 0 in Z.

According to (37) we can solve (36c) for p2 if and only if

(42)∫

Z

(A1p1 + A2p0) dz = 0.

Substituting (40) into (42) and considering Z-periodicity we find that

(43) ∇y ·(∫

Z

[A(∇yp0 +∇z

(u1

∂p0

∂y1+ u2

∂p0

∂y2

))]dz

)= 0.

This is identical to

(44) ∇y ·(A (x, y, z)A (x, y, z)

z∇yp0(x, y))

= 0,

where fz

=∫

Zf dz and

(45) A = A (x, y, z) =

⎛⎜⎝ 1 +∂u1

∂z1

∂u2

∂z1∂u1

∂z21 +

∂u2

∂z2

⎞⎟⎠ .

We remark that the equation (44) is the homogenized equation after the first reit-eration.

Equation (44) implies that

(46) p0(x, y) = p0(x).

Thus, by virtue of (40),

(47) p1 = p1 (x, y) .

Using (46) and (47) in (36c) and simplifying, we have

(48) A0p2 = ∇z · b−∇z · (A (∇yp1 +∇xp0)) .

By linearity we find that p2 is of the form

p2(x, y, z) = u0(x, y, z) + u1(x, y, z)(

∂p0

∂x1(x) +

∂p1

∂y1(x, y)

)(49)

+ u2(x, y, z)(

∂p0

∂x2(x) +

∂p1

∂y2(x, y)

)+ p2(x, y),

where u0 is a solution (unique up to an additive constant) to the local problem

(50) ∇z · (A∇zu0 − b) = 0 in Z.

Recall that even though y is a parameter in this context, u0 in (50) and u1 andu2 in (41) are not only Z-periodic, but also Y -periodic functions.


To solve (36d) for p3 it must hold that∫Z

(A1p2 + A2p1 + A3p0 −∇y · b) dz = 0.

Expansion yields

(51)∫

Z

(∇y · (A∇zp2) +∇y · (A∇yp1) +∇y · (A∇xp0)−∇y · b) dz = 0.

Inserting (49) in (51) and rearranging the terms we obtain that∫Z

∇y ·[A(∇xp0 +∇z

(u1

∂p0

∂x1+ u2

∂p0

∂x2

))+

+ (A∇zu0 − b) + A(∇yp1 +∇z

(u1

∂p1

∂y1+ u2

∂p1

∂y2

))]dz

= 0,

and by virtue of (45), this reduces to

(52) ∇y ·(AAz∇yp1

)= −∇y ·

(AAz∇xp0

)+∇y · (b−A∇zu0)

z.

By linearity, the equation (52) is satisfied if p1 is of the form

(53) p1(x, y) = v0(x, y) + v1(x, y)∂p0

∂x1(x) + v2(x, y)

∂p0

∂x2(x) + p1(x).

where the Y -periodic functions vi = vi (x, y) (i = 0, 1, 2) are the solutions of thefollowing local problems involving y

(54)

⎧⎨⎩ ∇y ·(AAz∇yv0 −

(b−A∇zu0

z))

= 0 in Y,

∇y ·(AAz

(∇yvi + ei))

= 0 on Y, (i = 1, 2) .

Here x, is regarded as a parameter.A necessary condition for solving (36e) for p4 is that

(55)∫

Z

A1p3 + A2p2 + A3p1 + A4p0 −∇x · b dz = 0.

By integrating (55) over Y , expanding the differential operators Ai and making useof the Y and Z periodicity yields

(56)∫

Y

∫Z

∇x · (A (∇zp2 +∇yp1 +∇xp0)− b) dydz = 0.

Next, we show that the condition (56) leads to the homogenized equation. Byinserting (49) and (53) in (56) we obtain

∇x ·

⎛⎝A∇z

(u0 + u1

(∂p0

∂x1+

∂p1

∂y1

)+ u2

(∂p0

∂x2+

∂p1

∂y2

)+ p2 (x, y)

)zy⎞⎠+

∇x ·

⎛⎝A∇y

(v0 + v1

∂p0

∂x1+ v2

∂p0

∂x2+ p1(x)

)zy⎞⎠+

∇x ·(A

zy

∇xp0

)= ∇x ·

(b

zy).


By simplifying and rearranging the following, we have that

∇x ·

⎛⎝A(∇xp0 +∇y

(v1

∂p0

∂x1+ v2

∂p0

∂x2

))zy⎞⎠+

∇x ·

⎛⎝A∇z

(u1

∂p0

∂x1+ u2

∂p0

∂x2

)zy⎞⎠+

∇x ·

⎛⎝A∇z

(u1

∂

∂y1

(v1

∂p0

∂x1+ v2

∂p0

∂x2

))zy⎞⎠+

∇x ·

⎛⎝A∇z

(u2

∂

∂y2

(v1

∂p0

∂x1+ v2

∂p0

∂x2

))zy⎞⎠ =

∇x ·

⎛⎝b−A(∇zu0 +∇yv0 +∇z

(u1

∂v0

∂y1

)+∇z

(u2

∂v0

∂y2

))zy⎞⎠ .(57)

By defining

(58) B = B (x, y) =

⎛⎜⎝ 1 +∂v1

∂y1

∂v2

∂y1∂v1

∂y21 +

∂v2

∂y2

⎞⎟⎠ ,

we see that the compressed form of

(59) ∇x ·(A(∇xp0 +∇y

(v1

∂p0

∂x1+ v2

∂p0

∂x2

)))= ∇x · (AB∇xp0) .

Inserting (59) into (57) and rearranging the terms, we find that

∇x ·(ABzy

∇xp0

)+

∇x ·

⎛⎜⎜⎜⎝A

⎛⎜⎝∂u1

∂z1

∂u2

∂z1∂u1

∂z2

∂u2

∂z2

⎞⎟⎠zy

∇xp0

⎞⎟⎟⎟⎠+

∇x ·

⎛⎜⎜⎜⎝A

⎛⎜⎝∂u1

∂z1

∂v1

∂y1

∂u1

∂z1

∂v2

∂y1∂u1

∂z2

∂v1

∂y1

∂u1

∂z2

∂v2

∂y1

⎞⎟⎠zy

∇xp0

⎞⎟⎟⎟⎠+

∇x ·

⎛⎜⎜⎜⎝A

⎛⎜⎝∂u2

∂z1

∂v1

∂y2

∂u2

∂z1

∂v2

∂y2∂u2

∂z2

∂v1

∂y2

∂u2

∂z2

∂v2

∂y2

⎞⎟⎠zy

∇xp0

⎞⎟⎟⎟⎠= ∇x ·

⎛⎝b−A(∇zu0 +∇yv0 +∇z

(u1

∂v0

∂y1

)+∇z

(u2

∂v0

∂y2

))zy⎞⎠ .


By adding the corresponding components of the matrices in the inner brackets andsimplifying, we obtain that

∇x ·(AABzy

∇xp0

)= ∇x ·

⎛⎝b−A∇zu0 −A(∇yv0 +∇zu1

∂v0

∂y1+∇zu2

∂v0

∂y2

)zy⎞⎠ .(60)

Moreover,

(61) ∇yv0 +∇zu1∂v0

∂y1+∇zu2

∂v0

∂y2= A∇yv0,

and thus from (61) and (60) we see that

(62) ∇x ·(AABzy

∇xp0

)= ∇x ·

(b−A∇zu0 −AA∇yv

z

0

y).

By defining

A0(x) = AABzy

,(63a)

b0(x) = b−A∇zu0 −AA∇yv0zy

,(63b)

and inserting in (62), we finally obtain that

∇x · (A0(x)∇xp0(x)) = ∇x · b0(x) in Ω,(64)

p0(x) = 0 on ∂Ω,

where A is defined as in (45) and B is defined as in (58). In other words, (64) is thereiterated homogenized boundary value problem corresponding to the deterministicboundary value problem given by (1).














Paper C

VARIATIONAL BOUNDS APPLIED TO UNSTATIONARYHYDRODYNAMIC LUBRICATION


Abstract. This paper is devoted to the effects of surface roughness in hydro-dynamic lubrication. The numerical analysis of such problems requires a veryfine mesh to resolve the surface roughness, hence it is often necessary to dosome type of averaging. Previously, homogenization (a rigorous form of aver-aging) has been successfully applied to Reynolds type differential equations.More recently, the idea of finding upper and lower bounds on the effective be-havior, obtained by homogenization, was applied for the first time in tribology.In these pioneering works, it has been assumed that only one surface is rough.In this paper we develop these results to include the unstationary case whereboth surfaces may be rough. More precisely, we first use multiple scale expan-sion to obtain a homogenization result for a class of variational problems in-cluding the variational formulation associated with the unstationary Reynoldsequation. Thereafter, we derive lower and upper bounds corresponding to thehomogenized (averaged) variational problem. The bounds reduce the numeri-cal analysis, in that one only needs to solve two smooth problems, i.e. no localscale has to be considered. Finally, we present several examples, where it isshown that the bounds can be used to estimate the effects of surface roughnesswith very high accuracy.

1. Introduction

It is well known that the surface micro topography is an important parameterin determining the performance of moving machine parts operating in the hydro-dynamic lubrication regime. An important problem in thin-film lubrication theoryis therefore to estimate the effects of surface roughness on the pressure solution.

In Reynolds lubrication model, the effects of surface roughness are solely deter-mined by the induced variations in film thickness, both in space and time. Due tothese oscillations, a direct computation of the pressure solution may not be pos-sible in practice since that would require a high resolution mesh. A remedy maybe to consider some sort of averaging. Surface roughness in hydrodynamic lubrica-tion has been considered by many authors and several averaging techniques havebeen proposed in the literature. A rigorous form of averaging is accomplished byhomogenization . Reynolds-type differential equations have been analyzed by ho-mogenization techniques in e.g. [1, 2, 3, 4, 5, 6, 7, 9, 11, 13] and by other averagingtechniques in e.g. [8, 10, 15, 16].

The procedure for solving the homogenized (averaged) equation can be describedas follows: First one solves a number of local problems by some numerical method.Then these local solutions are used to compute the coefficients in the homogenized

Key words and phrases. Unstationary Reynolds equation, homogenization, bounds, surfaceroughness.

1


equation. Finally, the homogenized equation is solved. This means that the ho-mogenized solution, although not as complex as the deterministic solution, maystill be very demanding to compute because of all the local problems that need tobe solved. In the recent works [2, 13], an alternative approach was introduced foranalyzing the stationary Reynolds equation (one rough surface). The advantageof this approach is that no local problems need to be computed. The main ideais to obtain bounds on the homogenized ”energy density” appearing in the varia-tional formulation corresonding to the homogenized equation. For a summary ofthe state of the art concerning bounds in general the reader may consult e.g. [12].The conclusion of [2, 13] was that when both precision and computational timeare important, bounds are a very cost-effective method of estimating the effects ofsurface roughness in stationary hydrodynamic lubrication.

The main purpose of this work is to develop the ideas in [2, 13] to include thesituation where both surfaces are rough. More precisely, in Section 2 we study thehomogenization of a class of variational problems by multiple scale expansion. Inparticular, this class of variational problems includes the variational formulationassociated with the stationary and the unstationary Reynolds equation formulatedin either Cartesian or polar coordinates. In Section 3 and 4 we derive boundsthat apply to the homogenized energy functional by using the techniques describedin [2, 13]. The bounds of arithmetic-harmonic mean type are optimal, i.e, thereexists surface roughness descriptions where the lower and upper bound coincide. InSection 5 we show that our general results may be applied to analyze the effects ofrough surface in connection with the unstationary Reynolds equation. Moreover,we present several numerical experiments. In particular, the pressure solutionsobtained by using the new bounds is compared with both the homogenized and thedeterministic solutions for different types of surface roughness. The experimentsclearly show that bounds may be used to accurately estimate the effects of surfaceroughness in hydrodynamic lubrication.

2. Homogenization of a variational principle

In this section we consider the homogenization of a class of variational problemsby multiple scale expansion. In particular, this class of variational problems in-cludes the variational formulation associated with the unstationary Reynolds equa-tion, which appears in hydrodynamic lubrication, see Section 4. The homogenizedvariational problem will thereafter serve as the starting point for deriving lower andupper bounds.

Let Ω be an open bounded subset of R2 and T > 0. In the remainder of thispaper ai, bi (i = 1, 2) and δ denote functions from Ω × [0, T ] × R

2 × R to R thatare Y -periodic in the third argument and Z-periodic in the fourth argument. Forsimplicity we assume Y = (0, 1)2 and Z = (0, 1). Moreover, define

A =(

a1 00 a2

)b = (b1, b2)

and assume that there exists a constant α such that a1, a2 ≥ α > 0.For each ε > 0 consider the variational principle

minp Iε(p)

Iε(p) def=∫ T

0

∫Ω

12∇xp ·Aε∇xp + bε · ∇xp + δεp dx dt

, (1)

VARIATIONAL BOUNDS 3

where

Aε(x, t) = A(x, t, x/ε, t/ε),bε(x, t) = b(x, t, x/ε, t/ε).δε(x, t) = δ(x, t, x/ε, t/ε)

A smooth functions p = p(x, t) satisfying p = 0 on the boundary of Ω is called anadmissible function and the minimum in (1) is understood to be taken over all suchfunctions p. Moreover, it is assumed that the minimization problem is well posedand we denote by pε the minimizer in the class of admissible functions.

As ε → 0 the coefficients in (1) will oscillate rapidly, which suggest some typeof averaging. The homogenization problem consists of analyzing the asymptoticbehavior as ε → 0. The main question is: does limε→0 Iε(pε) exist? We shallanswer this question in the affirmative by showing that there exist a homogenizedvariational problem minp I0(p), such that

limε→0

minp

Iε(p) = minp

I0(p).

We start the homogenization by first postulating that the sequence of real num-bers Iε(pε)ε>0 is bounded. Moreover, we assume that pε admits the followingmultiple-scale expansion

pε(x, t) =∞∑

i=0

εipi(x, t, x/ε, t/ε), (2)

where the functions pi = pi(x, t, y, τ) (i = 0, 1, 2, . . .) are assumed to be smoothfunctions that are Y -periodic in y and Z-periodic in τ . The idea is to apply thechain rule and formally insert (2) into (1) and to determine thereby the limitingbehavior of Iε(pε) as ε → 0. After some calculations we obtain the followingexpansion for Iε:

Iε(pε) = ε−2

∫ T

0

∫Ω

12∇yp0 ·Aε∇yp0 dx dt

+ ε−1

∫ T

0

∫Ω

∇yp0 ·Aε(∇xp0 +∇yp1) + bε · ∇yp0 dx dt

+ ε0

∫ T

0

∫Ω

12(∇xp0 +∇yp1) ·Aε(∇xp0 +∇yp1) +∇yp0 ·Aε(∇xp1 +∇yp2)

+ bε · (∇xp0 +∇yp1) + δεp0

dx dt + ε(. . . ). (3)

Consider the first term in the expansion (3), i.e.

ε−2

∫ T

0

∫Ω

12∇yp0 ·Aε∇yp0 dx dt.

In view of the conditions on ai we have that∫ T

0

∫Ω

12∇yp0 ·Aε∇yp0 dx dt ≥ α

2

∫ T

0

∫Ω

|∇yp0|2 dx dt


for some α > 0. Furthermore, according to the “mean value property” (see forexample [14])

limε→0

∫ T

0

∫Ω

|∇yp0(x, t, x/ε, t/ε)|2 dx dt

=∫ T

0

∫Ω

∫Z

∫Y

|∇yp0(x, t, y, τ)|2 dy dτ dx dt.

Thus, for the boundedness assumption to hold it is necessary that∫ T

0

∫Ω

∫Z

∫Y

|∇yp0|2 dy dτ dx dt = 0.

This implies ∇yp0 = 0, hence p0(x, t, y, τ) = p0(x, t, τ), and so (3) can be reducedto

Iε(pε) =∫ T

0

∫Ω

12(∇xp0 +∇yp1) ·Aε(∇xp0 +∇yp1)

+ bε · (∇xp0 +∇yp1) + δεp0

dx dt + O(ε). (4)

According to the mean value property we obtain

limε→0

Iε(pε) =∫ T

0

∫Ω

∫Z

∫Y

12(∇xp0 +∇yp1) ·A(∇xp0 +∇yp1)

+ b · (∇xp0 +∇yp1) + δp0 dy dτ dx dt. (5)

Thus, the limiting behavior of Iε(pε) is governed by the functions p0 and p1

alone. This suggests that p0 and p1 are determined by the variational principle⎧⎪⎨⎪⎩minp,w J0(p, w)

J0(p, w) def=∫ T

0

∫Ω

∫Z

∫Y

12 (∇xp +∇yw) ·A(∇xp +∇yw)

+b · (∇xp +∇yw) + δp dy dτ dx dt.

(6)

The minimum in (6) is taken over all smooth p = p(x, t, τ) that vanish on theboundary and smooth w = w(x, t, y, τ) that are Y -periodic in y. In other words,we claim

limε→0

Iε(pε) = J0(p0, p1) = minp,w

J0(p, w).

Let Wper denote the set of all Y -periodic smooth functions w = w(y). Then we canwrite (6) as

minp I0(p)

I0(p) def=∫ T

0

∫Ω

∫Z

f0(x, t, τ,∇xp) + δ0(x, t, τ)p dτ dx dt, (7)

where δ0(x, t, τ) =∫

Yδ(x, t, y, τ) dy and

f0(x, t, τ, ξ) = minw∈Wper

∫Y

12(ξ +∇yw) ·A(x, t, y, τ)(ξ +∇yw)

+ b(x, t, y, τ) · (ξ +∇yw) dy, (8)

for any ξ ∈ R2.Summing up we have derived the following homogenization result:

limε→0

minp

Iε(p) = limε→0

Iε(pε) = I0(p0) = minp

I0(p), (9)


where I0 is given by (7).We proceed by giving an alternative formula for f0 that is more suitable for

computation. From the definition of f0 we obtain

f0(x, t, τ, ξ)

= minw∈Wper

∫Y

12∇yw ·A∇yw + Aξ · ∇yw +

12ξ ·Aξ + b · (ξ +∇yw) dy. (10)

Omitting all constant terms it is clear that a minimizer wξ of (10) is also a solutionof

minw∈Wper

∫Y

12∇yw ·A∇yw + (Aξ + b) · ∇yw dy. (11)

The Euler–Lagrange equation corresponding to (11) is

∇y ·(A∇yw + (Aξ + b)

)= 0.

By linearity w can be written w = vξ + v0, where vξ and v0 are solutions of thelocal problems

∇y ·A(ξ +∇yvξ) = 0 in Y (12)

∇y · (A∇yv0 + b) = 0 in Y. (13)

Inserting w = vξ + v0 into (10) yields

f0(x, t, τ, ξ) =∫

Y

12(ξ +∇yw) ·A(ξ +∇yvξ) +

12(ξ +∇yw) · (A∇yv0 + b) dy.

Integration by parts together with (12) and (13) further gives

f0(x, t, τ, ξ) =∫

Y

12ξ ·A(ξ +∇yvξ) + ξ · (A∇yv0 + b)− 1

2∇yv0 ·A∇yv0 dy

=12ξ ·A0ξ + b0 · ξ +

12

∫Y

b · ∇yv0dy, (14)

where

A0ξ =∫

Y

A(ξ +∇yvξ) dy, b0 =∫

Y

(A∇yv0 + b) dy.

It can be noted from the homogenization result above that it is possible to deducea homogenization result for the Euler equation to (1), i.e.

−∇x · (Aε∇xpε + bε) + δε = 0 in Ω× (0, T ). (15)

By taking the alternative formula (14) and (7) into account it follows that thecorresponding homogenized equation is

−∇x · (A0∇xp0 + b0) + δ0 = 0 in Ω× (0, T ). (16)

We remark that the homogenization of (15) was studied in [3] by a different methodnamely two-scale convergence and that their results are in agreement with ours.


3. Bounds of Arithmetic-Harmonic Type

The aim of this section is to extend the ideas in [2, 13] in order to obtain upperand lower bounds on the function f0. First we note that x, t and τ may be regardedas parameters. To emphasize this and to avoid lengthy notation we temporarilywrite A = A(y) and b = b(y).

More precisely, we prove an estimate of the type

12ξ ·A−ξ + b− · ξ + c− ≤ f(ξ) ≤ 1

2ξ ·A+ξ + b+ · ξ + c+,

where A± (diagonal matrices), b± (vectors) and c± (scalars) are constants, on thefunction f : R2 → R defined by

f(ξ) = minw∈Wper

∫Y

12(ξ +∇yw) ·A(ξ +∇yw) + b · (ξ +∇yw) dy. (17)

The proof of the bounds now follows by using the same arguments as in [13], butis included for the readers convenience.

3.1. Upper bound. Let V = φ ∈ Wper : φ(y) = φ1(y1) + φ2(y2). Then clearly

f(ξ) ≤ minw∈V

∫Y

12(ξ +∇yw) ·A(ξ +∇yw) + b · (ξ +∇yw) dy

def= f+(ξ). (18)

We compute f+(ξ) explicitly by solving the corresponding weak Euler–Lagrangeequation: ∫

Y

(A(ξ +∇yw) + b) · ∇yφdy = 0 ∀φ ∈ V. (19)

Upon inserting w(y) = w1(y1)+w2(y2) and φ(y) = φ1(y1)+φ2(y2) into (19), weobtain∫ 1

0

∫ 1

0

(a1(ξ1 + w′1) + b1

)dy2

φ′1(y1) dy1

+∫ 1

0

∫ 1

0

(a2(ξ2 + w′2) + b2

)dy1

φ′2(y2) dy2 = 0.

We conclude that the expressions within curly brackets must be constant.

Notation. We assume in the sequel that the indices i and j are complementary in1, 2, i.e. either (i, j) = (1, 2) or (i, j) = (2, 1).

Thus, ki (i = 1, 2) defined by

ki =∫ 1

0

ai(y) (ξi + w′i(yi)) + bi(y) dyj . (20)

are constants. But (20) implies that

ξi + w′i =ki −Bi

Ai, (21)

where

Ai(yi) =∫ 1

0

ai(y) dyj and Bi(yi) =∫ 1

0

bi(y) dyj .


We compute ki by integrating (21) on (0, 1) with respect to yj and then solve forki. This gives

ki =ξi +

∫ 1

0BiAi

dyi∫ 1

0A−1

i dyi

.

After some straight forward computations we obtain

f+(ξ) =2∑

i=1

12

(ξi +

∫ 1

0BiAi

dyi

)2

∫ 1

0A−1

i dyi

− 12

∫ 1

0

B2i

Aidyi. (22)

Define the matrix A+ and the vector β+ as

A+(y) =

(∫ 1

0a1 dy2 00

∫ 1

0a2 dy1

)and β+(y) =

(∫ 1

0b1 dy2∫ 1

0b2 dy1

).

Then f+ can be written as

f+(ξ) =12(ξ +⟨(A+)−1β+

⟩)·⟨(A+)−1

⟩−1 (ξ +⟨(A+)−1β+

⟩)− 1

2⟨β+ ·(A+)−1β+

⟩,

where 〈f〉 denotes the integral of f over Y , i.e. 〈f〉 =∫

Yf dy. From this we see

thatf+(ξ) =

12ξ ·A+ξ + b+ · ξ + c+, (23)

where

A+ =⟨(A+)−1

⟩−1, b+ =

⟨(A+)−1

⟩−1⟨(A+)−1β+⟩

and (24)

c+ =12⟨(A+)−1β+

⟩·⟨(A+)−1

⟩−1⟨(A+)−1β+⟩− 1

2⟨β+ · (A+)−1β+

⟩. (25)

3.2. Lower bound. The upper bound f+ was straight forward to derive but alower bound calls for more sophisticated techniques. The derivation of lower boundsrelies on the dual variational principle (see Appendix A)

f∗(η) = minσ∈S

∫Y

12(σ + η − b) ·A−1(σ + η − b) dy, (26)

where f∗ : R2 → R denotes the conjugate function of f , or the Legendre transfor-mation of f , defined by

f∗(η) = maxξ∈R2

η · ξ − f(ξ)

and S = σ : Y → R2 : ∇ · σ = 0 and

∫Y

σ dy = 0 is the space of solenoidal vectorfields on Y with mean value zero. As before, the idea is to look for a solution ofthe minimization problem (26) in a smaller subspace.

Define the vector field space S∗ ⊂ S by

S∗ = σ ∈ S : σ = (σ1(y2), σ2(y1)).Obviously

f∗(η) ≤ minσ∈S∗

∫Y

12(σ + η − b) ·A−1(σ + η − b) dy

def= (f∗)+(η). (27)

We compute (f∗)+(η) by solving the corresponding Euler–Lagrange equations: findσ ∈ S∗ such that ∫

Y

(φ1 00 φ2

)A−1(σ + η − b) dy = 0 (28)


for all φ1, φ2 with mean value zero. From (28) we see that∫ 1

0

∫ 1

0

a−1i (σi(yj) + ηi − bi) dyi

φi(yj) dyj = 0.

By choosing φi = ψ′, such that ψ is smooth and has compact support in (0, 1), wecan conclude that the expression between the curly brackets does not depend on y.Thus, mi defined by

mi =∫ 1

0

a−1i (σi(yj) + ηi − bi) dyi (29)

is constant, where i and j are complementary. However, (29) implies

σi + ηi = Hi

(mi + Gi

), (30)

where

Hi =(∫ 1

0

a−1i dyi

)−1

Gi =∫ 1

0

bi

aidyi.

Integration of (30) w.r.t. yj yields

mi =ηi −

∫ 1

0HiGi dyj∫ 1

0Hi dyj

. (31)

By the fundamental theorem of calculus

(σi + ηi − bi)2ai

2

− b2i

2ai=∫ σi+ηi

0

(θ − bi)ai

dθ.

We compute

∫Y

∫ σi+ηi

0

(θ − bi)ai

dθ dy =

(ηi −

∫ 1

0HiGi dyj

)2

2∫ 1

0Hi dyj

− 12

∫ 1

0

HiG2i dyj

to find that∫Y

(σi + ηi − bi)2ai

2

dy =

(ηi −

∫ 1

0HiGi dyj

)2

2∫ 1

0Hi dyj

− 12

∫ 1

0

HiG2i dyj +

∫Y

b2i

2aidy

and consequently

(f∗)+(η) =2∑

i=1

(ηi −

∫ 1

0HiGi dyj

)2

2∫ 1

0Hi dyj

+∫

Y

b2i

2aidy − 1

2

∫ 1

0

HiG2i dyj .

Now define

A−(y) =

(∫ 1

0a−11 dy1 00

∫ 1

0a−12 dy2

)−1

=(

H1(y2) 00 H2(y1)

),

and

β−(y) = A−(y)(∫ 1

0b1a1

dy1,∫ 1

0b2a2

dy2

)=(H1G1(y2),H2G2(y1)

).

Then

(f∗)+(η) =12(η − 〈β−〉) · 〈A−〉−1(η − 〈β−〉) +

12〈b ·A−1b〉 − 1

2〈β− · (A−)−1β−〉.


Because of the involutive property of the Legendre transformation the conjugatefunction of (f∗)+, ((f∗)+)∗, yields a lower bound on f . We compute(

(f∗)+)∗(ξ) =

12ξ · 〈A−〉ξ + 〈β−〉 · ξ +

12〈β− · (A−)−1β−〉 − 1

2〈b ·A−1b〉.

Hence, f is bounded from below by the function

f−(ξ) =12ξ ·A−ξ + b− · ξ + c−, (32)

where

A− = 〈A−〉 b− = 〈β−〉 c− =12〈β− · (A−)−1β−〉 − 1

2〈b ·A−1b〉. (33)

It should be noted that the bounds of arithmetic-harmonic mean type are opti-mal, in the sense that there exists matrices A for which f−0 and f+

0 coincide withf0.

4. Bounds of Reuss–Voigt type

A trivial bound on f is obtained by taking w = 0 in the right hand side of (17).Indeed

f(ξ) ≤∫

Y

12ξ ·Aξ + b · ξ dy =

12ξ · 〈A〉ξ + 〈b〉 · ξ def= f+

RV(ξ).

Similarly, by taking σ = 0 in the dual variational principle (26), it is clear that

f∗(η) ≤∫

Y

12(η − b) ·A−1(η − b) dy

=12η · 〈A−1〉η − η · 〈A−1b〉+

12〈b ·A−1b〉.

The Legendre transformation of the last expression is

f−RV(ξ) def=12(ξ + 〈A−1b〉) · 〈A−1〉−1(ξ + 〈A−1b〉)− 1

2〈b ·A−1b〉,

which yields the lower Reuss–Voigt bound on f . This proves the pointwise estimatef−RV ≤ f ≤ f+

RV; f±RV(ξ) = 12ξ ·A±RVξ + b±RV · ξ + c±RV where

A+RV = 〈A〉 A−RV = 〈A−1〉−1 (34a)

b+RV = 〈b〉 b−RV = 〈A−1〉−1〈A−1b〉 (34b)

c+RV = 0 c−RV =

12〈A−1b〉 · 〈A−1〉−1〈A−1b〉 − 1

2〈b ·A−1b〉. (34c)

5. Application to a problem in hydrodynamic lubrication

We apply the preceding general results to a thrust pad bearing problem whereboth the pad and the shaft surfaces exhibit periodic roughness and the lubricantis assumed to be an incompressible Newtonian fluid with viscosity μ and densityρ. For a schematic description of the model problem see Figure 1. A point on thereference plane is identified by its polar coordinates x = (x1, x2), x1 denoting theangular coordinate and x2 the radial coordinate. The gap between the two smoothsurfaces is given by h0 and the lower surface rotates uniformly at the angular speedω.

To model surface roughness we introduce the auxiliary film thickness function

h(x, y, τ) = h0(x) + hU(y)− hL(y − τ(ω, 0)), (35)


where• h0 is a function of the global variable x (space variable) that describes the

global shape of the film thickness,• hU and hL are Y -periodic functions of the local variables y = x/ε and

τ = t/ε that represent the roughness contribution of the upper and lowersurfaces respectively.

Note that the results obtained in Section 2 are easily extended to allow for anarbitrary ω by defining Z = (0, 1/ω).

For a given wavelength ε, the corresponding film thickness hε is given by hε(x, t) =h(x, x/ε, t/ε). The pressure pε = pε(x, t) that builds up in the film is assumed tobe governed by Reynolds variational principle, i.e. Iε(pε) = minp Iε(p) where

Iε(p) =∫ T

0

∫Ω

h3ε

2x2

∣∣∣∣ ∂p

∂x1

∣∣∣∣2 +x2h

3ε

2

∣∣∣∣ ∂p

∂x2

∣∣∣∣2 − λx2h∂p

∂x1dx dt, (36)

h is defined by h = h0 + hU + hL and γ = 12μ and λ = γω/2 are constants.Moreover, the bearing domain is assumed to be annulus-shaped, i.e. there existsR1 and R2 such that for each x ∈ Ω it holds that 0 < R1 ≤ x2 ≤ R2.

h0(x)

x1, ω

x2

Figure 1. Schematic of a thrust pad bearing.

To see that (36) is of the general form (1), take

A(x, y, τ) =(

a1(x, y, τ) 00 a2(x, y, τ)

)= h(x, y, τ)3

(1/x2 0

0 x2

),

b(x, y, τ) = (b1(x, y, τ), 0) = −λx2h(x, y, τ)(1, 0),δ0(x, t) = 0.

Since h is of special form (35), it is readily checked that

λ∂hε

∂x1(x, t) + γ

∂hε

∂t(x, t) = λ

∂

∂x1h(x, x/ε, t/ε).

Using this relation we see that the Euler–Lagrange equation corresponding to (36)is the unstationary Reynolds equation in polar coordinates, i.e.

∂

∂x1

(h3

ε

x2

∂pε

∂x1

)+

∂

∂x2

(x2h

3ε

∂pε

∂x2

)= λx2

∂hε

∂x1+ γx2

∂hε

∂t.


The coefficients in the arithmetic-harmonic and Reuss–Voigt bounds are easilycomputed according to the formulae (25), (33) and (34) respectively. By using thebounds (23) and (32) we obtain

I−0 (p−) = minp

I−0 (p) ≤ minp

I0(p) ≤ minp

I+0 (p) = I+

0 (p+),

where

I±0 (p) =∫ T

0

∫Ω

∫Z

12∇xp ·A±∇xp + b± · ∇xp + c± dτ dx dt. (37)

The Euler–Lagrange equations corresponding to I±0 are

−∇x ·(A±∇xp± + b±

)= 0 in Ω× (0, T ), (38)

which allows us to compute the bounds solutions p+ and p−.We now proceed with some numerical examples. All results are presented in

dimensionless form, i.e. the following dimensionless variables are introduced:

X2 =x2

R1, H =

h

h00, P = λ

h200

R21

p,

where the inner pad radius R1 and h00, the film thickness at the trailing edge (ofh0), are used as scaling parameters. In this case, the dimensionless global filmthickness H0 is given by

H0 (X) = 1−KX2R1

R2

sin X1 − sin θ2

sin θ2 − sin θ1,

where R1/R2 = 3/7, θ2 = −θ1 = 27.5 and K = 1/4. These values are used in thesimulation of a single pad in a thrust pad bearing, which is assumed to consist ofa total of 6 pads separated by an angle of 5 and operating at 1/4 inclination withR1/R2 = 3/7. This leads to the dimensionless Reynolds equation:

∇X ·(

H3ε

(1/X2 0

0 X2

)∇XPε

)= X2

∂Hε

∂X1. (39)

The advantage of this form is that it does not contain any reference or input pa-rameters. By solving this equation once, for a given R1/R2 ratio, a given constantK and for a specific surface roughness representation (hU and hL), we simulatea 6 pad bearing, with pads separated by an angle of 5, given any choice of theparameters μ, h00, R1 and ω (if the generalized Z = (0, 1/ω) is considered). In thesubsequent sections we present some illustrative cases of the numerical simulationsperformed.

5.1. Sinusoidal roughness. First, roughness due to a one-dimensional transversalsinusoidal perturbation of the global film thickness is considered. More precisely,the roughness is described by

HU (y, τ) = − c

2(sin (2πy1)− 1) ,

HL (y, τ) =c

2(sin (2π (y1 − ωτ))− 1) ,

(40)

where c = 1/8. Figure 2 is a plot of the dimensionless deterministic pressuredistribution corresponding to the perturbations (40) of the global film H0. Aspointed out before, the bounds coincide for transversal roughness, hence P− =P0 = P+ for sinusoidal roughness. In general, the computation of the homogenizedsolution is a fairly complex task and several local problems need to be solved in


0.10.15

0.20.25

0.30.35

0.20.1

00.1

0.20

0.5

1

1.5

2

2.5

3x 10 3

x1 (m)x

2 (m)

P ε ()

Figure 2. Sinusoidal roughness: The pressure distribution Pε,ε = 1/24, at an intermediate time step.

the process. Even if it is not needed in this example, we use this general approachto compute P0. To quantify the numerical accuracy, the following measure of therelative difference between the bounds solutions and the homogenized solution isintroduced ∫

Ω

∣∣P± (x, τk)− P0 (x, τk)∣∣ dx/∫

Ω

P± (x, τk) dx,

Figure 3 represents a plot of this measure against τk for an eight-step overtakingcycle. Recall that the solutions P0 and P± are functions of x, τ and periodic inτ , whereas the deterministic solution Pε is a function of x, t and periodic in t.In this simulation a single period of the sinusoids was discretely represented by64 spatial nodes. According to Figure 3 the maximum relative difference did notexceed 0.02%, which verifies the feasibility, regarding the numerical accuracy, ofour method for computing P0 by the general procedure.

The deterministic load carrying capacity lccε(t) =∫Ω

Pε (x, t) dx and the loadcarrying capacity lcc0(τ) =

∫Ω

P0 (x, τ) dx associated with the homogenized solu-tion are depicted in Figure 4. It is clear that for ε = 1/26 the difference betweenthe load carrying capacities associated with the deterministic solution Pε and thehomogenized solution P0 is small. In fact, the difference attains a maximum valuewhich is less than 2.5%, half way through the cycle (τ = τ4).

In Figure 5 three pressure solutions Pε, ε = 1/24, 1/25 and 1/26 - at an interme-diate time step and x2 = Rm, where Rm = (1 + R2/R1)/2 - are shown. For clarityan enlarged portion of the central region is also displayed (Right). In Figure 6 anenlarged portion of the deterministic pressure solution Pε, for ε = 1/26 and thehomogenized solution P0 at three consecutive time steps is shown. In particular,this figure illustrates the unstationary behavior of Reynolds equation. Finally, theReuss–Voigt bounds are compared to the homogenized solution. Figure 7 shows asnap shot of the Reuss–Voigt bounds pressure solutions P±

RV and the homogenizedsolution P0 at an intermediate time step and Figure 8 the measure of the rela-tive differences

∫Ω

∣∣P±RV (x, τk)− P0 (x, τk)

∣∣ dx/∫

ΩP0 (x, τk) dx , for the complete


0 1 2 3 4 5 6 70.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 x 10 5

Discrete τ step

Rel

ativ

e di

ffer

ence

Figure 3. Sinusoidal roughness: A τ -cycle of the relative dif-ference in load carrying capacity between the aritmetic-harmonicbounds solution and the homogenized solution.

0 1 2 3 4 5 6 72.25

2.3

2.35

2.4

2.45

2.5

2.55 x 10 3

Discrete t / τ step

Load

car

ryin

g ca

paci

ty

ε=1/24

ε=1/25

ε=1/26

hom.

Figure 4. Sinusoidal roughness: A cycle of the deterministic loadcarrying capacity and homogenized dito computed at each timestep for three different ε.

τ -cycle. Of course, the preciseness of the Reuss–Voigt mean type bounds is notcomparable to that of the coinciding arithmetic-harmonic bounds, however, as canbe observed from Figure 8 the maximum relative difference between the upper andthe lower bound is less than 5%.


30 20 10 0 10 20 300

0.5

1

1.5

2

2.5

3x 10 3

x1 (deg.)

Dim

ensi

onle

ss P

ress

ure

1 2 3 4 52.3

2.4

2.5

2.6

2.7x 10 3

x1 (deg.)

Figure 5. Sinusoidal roughness: The pressure solution at x2 =Rm, for three different choices of ε (Left). An enlargement showingthe central region. The homogenized solution is indicated by thedashed line. (Right).

2 2.5 3 3.5 4

2.4

2.45

2.5

2.55

2.6 x 10 3

x1 (deg.)

Dim

ensi

onle

ss P

ress

ure

Pε(x1,Rm,t2)

P0(x1,Rm,τ2)

Pε(x1,Rm,t3)

P0(x1,Rm,τ3)

Pε(x1,Rm,t4)

P0(x1,Rm,τ4)

Figure 6. Sinusoidal roughness: The deterministic pressure so-lution Pε, ε = 1/26, and the homogenized solution P0 at threeconsecutive time steps.

5.2. Bisinusoidal roughness. The second simulation is devoted to bisinusoidalsurface roughness, i.e., roughness in both directions and we define HU and HL as

HU (y, τ) = − c

2(cos (2πy1) cos (2πy2)− 1) ,

HL (y, τ) =c

2(cos (2π (y1 − ωτ)) cos (2πy2)− 1) .

(41)


2 2.5 3 3.5 4

2.4

2.45

2.5

2.55

2.6 x 10 3

x1 (deg.)

Dim

ensi

onle

ss P

ress

ure

PRV(x1,Rm,τ2)

P0(x1,Rm,τ2)

P+RV(x1,Rm,τ2)

Figure 7. Sinusoidal roughness: The Reuss–Voigt bounds pres-sure solutions P±

RV and the homogenized solution P0 at τ = τ2.

0 1 2 3 4 5 6 70

0.005

0.01

0.015

0.02

0.025

0.03

Discrete τ step

Rel

ativ

e di

ffer

ence

Figure 8. Sinusoidal roughness: A τ -cycle of the relative differ-ence in load carrying capacity between the Reuss–Voigt bounds so-lutions and the homogenized solution. The upper bound solutionis indicated by filled squares, the lower bound solution is indicatedby filled circles.

As before, c = 1/8. The pressure distribution Pε, ε = 1/24, is shown in Figure 9and the convergence of the deterministic solution is investigated in Figure 10 wherethe load carrying capacity is plotted as a function of τ . We note that each period of


0.10.15

0.20.25

0.30.35

0.20.1

00.1

0.20

0.5

1

1.5

2

2.5

3x 10 3

x1 (m)x

2 (m)

P ε ()

Figure 9. Bisinusoidal roughness: The pressure distribution Pε,ε = 1/24, at an intermediate time step.

0 1 2 3 4 5 6 72.528

2.53

2.532

2.534

2.536

2.538

2.54 x 10 3

Discrete t / τ step

Load

car

ryin

g ca

paci

ty

ε=1/24

ε=1/25

ε=1/26

hom.

Figure 10. Bisinusoidal roughness: τ -cycles of homogenized anddeterministic load carrying capacities, for three choices of ε.

the deterministic, bisinusoidal, roughness representation is resolved with only 8× 8discrete grid nodes - meaning a total number of 513× 513 grid nodes for ε = 1/26.For this type of roughness the arithmetic-harmonic bounds pressure solutions arenot equal to the homogenized one but Figure 11 reveals that the difference betweenthe load carrying capacity corresponding to the upper and the lower bound is small.In the bisinusoidal case, we also observe that the Reuss–Voigt bounds are close -the difference between the upper and the lower bound is almost as small as for thearithmetic-harmonic bounds - with the maximum difference being approximately


0 1 2 3 4 5 6 70

0.002

0.004

0.006

0.008

0.01

0.012

Discrete τ step

Rel

ativ

e di

ffer

ence

Figure 11. Bisinusoidal roughness: A τ -cycle of the relative dif-ference in load carrying capacity between the arithmetic-harmonicbounds solutions. The upper bound solution is indicated by filledsquares, lower bound is indicated by filled circles.

2.40% as compared to (≈ 1.03 + 1.35 =) 2.38% for the arithmetic-harmonic bounds.For details see [2].

5.3. A realistic surface roughness representation. Figure 12 shows a surface

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

0.5

1

y1 ( )

y2 ( )

f r ()

Figure 12. A surface roughness representation originating froma real surface measurement.

originating from an optical interference measurement. The original grounded sur-face was coarsened by resampling on a 17 × 33 grid - to reduce the discretizationerrors by enabling successive linear interpolation - and also normalized. For the


results presented here each period was interpolated onto 65 × 129 discrete nodes.Let the roughness of the measured specimen be represented by the function fr,which is extended by periodicity. As we want to state the governing equation indimensionless form, let

HU (y, τ) = −c (f (y)− 1) ,

HL (y, τ) = c (f (y1 − ωτ, y2)− 1) ,(42)

where, as before, c = 1/8. Solving the deterministic problem, corresponding to thesurface roughness representation (42) with fr as in Figure 12 yields the pressuredistribution, at an intermediate time step tk as shown in Figure 13. Figure 14

0.10.15

0.20.25

0.30.35

0.20.1

00.1

0.20

0.5

1

1.5

2

2.5

3x 10 3

x1 (m)x

2 (m)

P ε ()

Figure 13. Realistic roughness: The pressure distribution, Pε,ε = 1/23, at an intermediate time step.

displays the deterministic and the Reuss–Voigt bounds pressure solutions, for thesame time step as in Figure 13, at x2 = Rm. Figure 15 is a magnification ofthe central region, where also the arithmetic-harmonic bounds have been added.This figure highlights the preciseness of the bounds as it visualizes the very smalldifferences between the upper and the lower bounds pressure solutions. To be moreprecise, the relative difference between the upper and lower arithmetic-harmonicbound is only about 0.1%. A small investigation was carried out showing thatwhen the amplitude of the surface roughness increase by a factor of 2 the differencebetween the upper and lower bound solution increase by a factor slightly less than4 at all τ -steps. Note that this also holds for the Reuss–Voigt bounds as well asfor the previously considered sinusoidal and bisinusoidal problems. For example,considering the arithmetic-harmonic bounds, c = 1/4, 1/2 and 1 yields relativedifferences of approximately (and certainly less than) 0.4%, 1.6% and 3.2%, for therealistic surface representation (42) with fr as in Figure 12.

6. Conclusions

It is well-known that homogenization is very useful in analyzing the effects ofsurface roughness on the performance of moving machine parts operating in the hy-drodynamic lubrication regime. The results presented here confirm this once again.


30 20 10 0 10 20 300

0.5

1

1.5

2

2.5

3 x 10 3

x1 (deg.)

Dim

ensi

onle

ss P

ress

ure

PRVP

ε

P+RV

Figure 14. Realistic roughness: The deterministic and theReuss–Voigt pressure solutions at x2 = Rm, same time step asin Figure 13

1 1.5 2 2.5 3 3.5 4 4.5 52.38

2.39

2.4

2.41

2.42

2.43

2.44

2.45

2.46

2.47

2.48 x 10 3

x1 (deg.)

Dim

ensi

onle

ss P

ress

ure

PRV

PP

ε

P+

PRV+

Figure 15. Realistic roughness: An enlargement of the centralregion of Figure 14 with also the arithmetic-harmonic boundsadded.

However, the homogenized solution has a severe limitation in that it may be verydemanding to compute because of all the local problems that need to be solved.In the recent works [2, 13], an alternative approach was introduced in connectionwith problems which are modeled by the stationary Reynolds equation (one rough


surface). The main result of this work is that these ideas have been extended toinclude the situation where both surfaces are rough. In order to do so we first de-rived a homogenization result for a class of variational problems, using the methodof multiple scale expansion. Thereafter, we proved bounds on the corresponding ho-mogenized local “energy density”. We showed that the class of bounds includes e.g.those associated with problems which are modeled by the unstationary Reynoldsequation (both surfaces rough) formulated in either Cartesian or polar coordinates.The bounds are optimal, i.e. there exists surface roughness descriptions where thelower and upper bound coincide. The advantage of using bounds instead of the ex-plicit formulas obtained by homogenization is that the complexity of the numericalanalysis is significantly reduced because one must solve only two smooth problems,i.e. no local scale has to be considered. To demonstrate that the bounds may beused to accurately estimate the effects of surface roughness in hydrodynamic lubri-cation, we have presented several numerical examples in connection with problemsthat are modeled by the unstationary Reynolds equation.

References

[1] A. Almqvist, E. K. Essel, L.-E. Persson, and Peter Wall. Homogenization of the unstationaryincompressible Reynolds equation. Tribology International, 40:1344–1350, 2007.

[2] A. Almqvist, D. Lukkassen, A. Meidell, and P. Wall. New concepts of homogenization appliedin rough surface hydrodynamic lubrication. Int. J. Eng. Sci., 45:139–154, 2007.

[3] G. Bayada, S. Ciuperca, and M. Jai. Homogenization of variational equations and inequal-ities with small oscillating parameters. Application to the study of thin film unstationarylubrication flow. C. R. Acad. Sci. Paris, t. 328, Serie II b:819–824, 2000.

[4] G. C. Buscaglia, I. Ciuperca, and M. Jai. Homogenization of the transient Reynolds equation.Asymptotic Analysis, 32(2):131–152, Nov 2002.

[5] G. C. Buscaglia and M. Jai. Sensitivity analysis and Taylor expansions in numerical homog-enization problems. Numerische Mathematik, 85:49–75, 2000.

[6] G. C. Buscaglia and M. Jai. A new numerical scheme for non uniform homogenized problems:Application to the non linear Reynolds compressible equation. Mathematical Problems inEngineering, 7:355–378, 2001.

[7] G. C. Buscaglia and M. Jai. Homogenization of the generalized Reynolds equation for ultra-thin gas films and its resolution by FEM. Journal of Tribology, 126(3):547–552, jul 2004.

[8] H. Christensen and K. Tonder. The hydrodynamic lubrication of rough bearing surfaces offinite width. ASME J. Lubr. Technol., 93:324–330, 1971.

[9] H. G. Elrod. Thin-film lubrication theory for newtonian fluids with surfaces possessing striatedroughness or grooving. ASME Journal of Lubrication Technology, 95:484–489, oct 1973.

[10] S. R. Harp and R. F. Salant. An average flow model of rough surface lubrication with inter-asperity cavitation. Journal of Tribology, 123(1):134–143, 2001.

[11] M. Jai and B. Bou-Said. A comparison of homogenization and averaging techniques for thetreatment of roughness in slip-flow-modified Reynolds equation. Transactions of the ASME.Journal of Tribology, 124(2):327–335, April 2002.

[12] V.V. Jikov, S.M. Kozlov, and O.A. Oleinik. Homogenization of Differential Operators andIntegral Functionals. Springer-Verlag, Berlin-Heidelberg-New York, 1994.

[13] D. Lukkassen, A. Meidell, and P. Wall. Bounds on the effective behavior of a homogenizedReynolds-type equation. Journal of Function Spaces and Applications, 5(2):133–150, 2007.

[14] D. Lukkassen, G. Nguetseng, and P. Wall. Two-scale convergence. Int. J. Pure Appl. Math.,2:35–86, 2002.

[15] Y. Mitsuya. A simulation method for hydrodynamic lubrication of surfaces with two-dimensional isotropic or anisotropic roughness using mixed averaging film thickness. Bulletinof JSME, 231(27):2036–2044, 1984.

[16] N. Patir and H. S. Cheng. An average flow model for determining effects of three-dimensionalroughness on partial hydrodynamic lubrication. Proc. 5th Leeds-Lyon Symp. Tribol., Lyon,pages 15–21, 1978.


Appendix A. A dual variational principle

Suppose Q is a symmetric invertible 2 × 2 matrix. As a special case of Young–Fenchel’s inequality the following holds:

12u ·Qu +

12v ·Q−1v ≥ u · v ∀u, v ∈ R2. (43)

Using (43) pointwise with Q = A(y), u = ξ +∇w(y) and v = σ(y) − b(y), whereσ is any vector field on Y , we obtain

f(ξ) ≥ minw∈Wper

∫Y

σ · (ξ +∇w)− 12(σ − b) ·A−1(σ − b) dy. (44)

Let Vsol consist of the Y -periodic vector fields that have zero divergence. Then itfollows from (44) that

f(ξ) ≥ maxσ∈Vsol

∫Y

σ · ξ − 12(σ − b) ·A−1(σ − b) dy. (45)

It turns out that the inequality (45) is actually an equality. Indeed, the solutionwξ of

minw∈Wper

∫Y

12(ξ +∇w) ·A(ξ +∇w) + b · (ξ +∇w) dy

is also the unique wξ in Wper such that∫Y

(A(ξ +∇wξ) + b) · ∇φdy = 0 ∀φ ∈ Wper. (46)

Let σ∗ = A(ξ+∇wξ)+b. From (46) it is clear that σ∗ belongs to Vsol. By choosingσ = σ∗ in (45) we obtain equality in (45), i.e.

f(ξ) = maxσ∈Vsol

∫Y

σ · ξ − 12(σ − b) ·A−1(σ − b) dy.

From the (orthogonal) decomposition Vsol = R2 ⊕ S, where S denotes the vector

fields in Vsol with mean value zero we have

f(ξ) = maxσ∈Vsol

∫Y

σ · ξ − 12(σ − b) ·A−1(σ − b) dy

= maxη∈R2

σ∈S

∫Y

η · ξ − 12(σ + η − b) ·A−1(σ + η − b) dy

= maxη∈R2

η · ξ −min

σ∈S

∫Y

12(σ + η − b) ·A−1(σ + η − b) dy

,

which shows that f is the Legendre transformation (w.r.t. the variable η) of thefunction

F (η) = minσ∈S

∫Y

12(σ + η − b) ·A−1(σ + η − b) dy.

In other words f = F ∗. Since F is convex and lower semicontinuous, (·)∗∗ acts asthe identity. Thus f∗ = F ∗∗ = F .

This proves the dual variational principle

f∗(η) = minσ∈S

∫Y

12(σ + η − b) ·A−1(σ + η − b) dy. (47)















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Homogenization Theory with Applications in Tribology