Homogenization of some problems in hydrodynamic lubrication involving rough boundaries

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(1)DOC TOR A L T H E S I S. ISSN: 1402-1544 ISBN 978-91-7439-254-8 Luleå University of Technology 2011. John Fabricius Homogenization of Some Problems in Hydrodynamic Lubrication Involving Rough Boundaries. Department of Engineering Sciences and Mathematics. Homogenization of Some Problems in Hydrodynamic Lubrication Involving Rough Boundaries. John Fabricius.

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(3) Homogenization of some problems in hydrodynamic lubrication involving rough boundaries John Fabricius Department of Engineering Sciences and Mathematics Lule˚ a University of Technology SE-971 87 Lule˚ a, Sweden.

(4) 2000 Mathematics Subject Classification. 35B27, 76D08 Key words and phrases. mathematics, partial differential equations, calculus of variations, homogenization theory, tribology, hydrodynamic lubrication, thin film flows, Reynolds equation, surface roughness, Weyl decomposition. Printed by Universitetstryckeriet, Luleå 2011 ISSN: 1402-1544 ISBN 978-91-7439-254-8 Luleå 2011 www.ltu.se.

(5) Abstract This thesis is devoted to the study of some homogenization problems with applications in lubrication theory. It consists of an introduction, five research papers (I–V) and a complementary appendix. Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficients. Many important problems in physics with one or several microscopic scales give rise to this kind of equations, whence the need for methods that enable an efficient treatment of such problems. To this end several mathematical techniques have been devised. The main homogenization method used in this thesis is called multiscale convergence. It is a notion of weak convergence in Lp spaces which is designed to take oscillations into account. In paper II we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with a finite number of microscopic scales. The main idea in the proof is closely related to a decomposition of vector fields due to Hermann Weyl. The Weyl decomposition is further explored in paper III. Lubrication theory is devoted to the study of fluid flows in thin domains. More generally, tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. The mathematical foundations of lubrication theory are given by the Navier–Stokes equation which describes the motion of a viscous fluid. In thin domains several simplifications are possible, as shown in the introduction of this thesis. The resulting equation is named after Osborne Reynolds and is much simpler to analyze than the Navier–Stokes equation. The Reynolds equation is widely used by engineers today. For extremely thin films, it is well-known that the surface micro-topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelengths upon the solution of the Reynolds equation. Since. iii.

(6) iv. ABSTRACT. the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are “averaged out”. One problem consists of finding an algorithm for computing the solution of the homogenized equation. Another problem consists of showing, on introducing the appropriate mathematical definitions, that the homogenized equation is the correct method of averaging. Papers I, II, IV and V investigate the effects of surface roughness by homogenization techniques in various situations of hydrodynamic lubrication. To compare the homogenized solution with the solution of the deterministic Reynolds equation, some numerical examples are also included..

(7) List of publications This thesis is composed of five papers (I–V) and a complementary appendix. These publications are put to a more general frame in an introduction that also serves as a basic overview of the field. I A. Almqvist, J. Fabricius and P. Wall. Homogenization of a Reynolds equation describing compressible flow. Research Report, No. 3, ISSN:1400-4003, Department of Engineering Sciences and Mathematics, Lule˚ a University of Technology, (24 pages), 2011. II A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Multiscale homogenization of a class of nonlinear equations with applications in lubrication theory. Journal of Function Spaces and Applications, 9(1):17–40, 2011. With an appendix: J. Fabricius. Analysis on the torus (8 pages). III J. Fabricius. On Weyl decomposition of vector fields. Research Report, No. 4, ISSN:1400-4003, Department of Engineering Sciences and Mathematics, Lule˚ a University of Technology, (26 pages), 2011. IV A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Reiterated homogenization applied in hydrodynamic lubrication. Proc. IMechE, Part J: J. Engineering Tribology, 222(7):827– 841, 2008. V A. Almqvist, E. K. Essel, J. Fabricius and P. Wall. Variational bounds applied to unstationary hydrodynamic lubrication. Internat. J. Engrg. Sci., 46(9):891–906, 2008.. v.

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(9) Acknowledgment I am grateful to my main advisors, Professor Lars-Erik Persson and Professor Peter Wall, for their constant support and for encouraging me to pursue my own research ideas. I thank my assistant advisor Docent Andreas Almqvist and co-author Ph.D. Emmanuel K. Essel for “frictionless” and fruitful collaboration. I have profited from the comments, thoughtful suggestions and valuable assistance of many of my colleagues and friends. In particular, I am grateful to Professor Sten Kaijser, Docent Thomas Str¨ omberg, Ph.D. Klas Pettersson, Ph.D. Yulia Koroleva, Ph.D. Johan Bystr¨ om and Elisabeth Andersson.. vii.

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(11) Introduction 1. Hydrodynamic lubrication Lubrication is the action of viscous fluids to diminish friction and wear between solid surfaces. It is fundamental to the operation of all engineering machines and many biological mechanisms. It can be observed that a converging fluid film is able to separate two bodies in relative motion pressed together under an external load. Thus, solid to solid contact is prevented and the applied load is supported by a pressure that develops within the film due to the fluid’s resistance to motion. Lubrication theory is devoted to the study of thin-film flows. The fundamental problem of lubrication can be stated as follows. Consider a viscous fluid (lubricant) which is brought into motion by two rigid bodies. The motions of the bodies are assumed to be known and the problem is to determine the dynamic behavior of the fluid—in particular the forces exerted by the fluid on the rigid surfaces. In general this problem is difficult to solve, but if the fluid film is “thin” one can show that the dominant part of the flow is governed by a simpler equation called the Reynolds equation. Starting from fairly general assumptions on the motions and the geometries of the rigid bodies, we present the solution of the lubrication problem in Theorem 9.1. 2. Fundamentals of fluid dynamics The compressible Navier–Stokes system describing viscous flow reads ∂u (1) + ρ(u · ∇)u = −∇p + (λ + µ)∇(div u) + µ∆u ρ ∂t ∂ρ (2) + div(ρu) = 0 ∂t where u = (u, v, w) is the unknown velocity field, ρ is the density of the fluid, p is the pressure and λ, µ are constants of viscosity. The system (1) is called the equation of motion. Equation (2) is the so called continuity equation which is equivalent to conservation of mass. Equations (1) and (2) are derived from first principles in Serrin [58]. If div u = 0, the flow 1.

(12) 2. INTRODUCTION. is said to be incompressible or volume preserving. This is the case if ρ is constant throughout the fluid. In hydrodynamic lubrication it may not be reasonable to assume that ρ is constant. Somewhat incorrectly, one refers to this as the compressible case. In this case one can assume that density is a function of pressure, i.e. ρ = ρ(p), determined by the compressibilty factor β of the fluid according to (3). dρ = βρ. dp. In general β may depend on both pressure and temperature. The stress tensor T for a Newtonian fluid (see [58, p. 236, eq. (61.1)]) is defined through the constitutive relation T = (−p + λ trace D)I + 2µD, where D = (∇u+(∇u)T )/2 is the deformation tensor (note that trace D = div u). Let Σ be a bounding surface of the fluid. According to Cauchy’s stress principle, the force f exerted by the fluid on the surface Σ is given by Z (4) f= T n dS Σ. where n denotes the unit normal to Σ pointing into the fluid domain and dS denotes surface measure. 3. Geometry and kinematics of rigid bodies The domain of the lubrication problem is rather complicated because the film will depend on both the surface geometry of the rigid bodies and their relative motion. In order to properly define the space-time domain of our problem, we need to recall some basic facts concerning geometry and kinematics of rigid bodies (for more details we refer to Arnold [7, Ch. 6] and Lax [39, Ch. 11]). A machine element, such as a piston head, journal or bearing pad, is an example of a rigid body. In mathematical language, a rigid body is a set of points p = (x, y, z) in three-dimensional Euclidian space R3 . Here we shall define a rigid body K as a compact three-dimensional manifold with boundary. This corresponds well to our intuitive notion of the examples given above. The boundary of K is denoted by ∂K and is by assumption a two-dimensional closed manifold, i.e. ∂(∂K) = ∅. As an orientation for K we choose the orientation for R3 corresponding to the ordered standard basis (e1 , e2 , e3 ), where e1 = (1, 0, 0),.

(13) 3. GEOMETRY AND KINEMATICS OF RIGID BODIES. 3. e2 = (0, 1, 0) and e3 = (0, 0, 1). An orientation on K induces an orientation on ∂K which can be defined in the following way (see Agricola– Friedrichs [1, p. 74] or Spivak [59, p. 119]). Let Tp M denote the tangent space of a manifold M at the point p. At each boundary point p ∈ ∂K, there exists a unique tangent vector n(p) ∈ Tp K such that (1) n(p) is perpendicular to Tp (∂K) (2) |n(p)| = 1 (3) for each coordinate system ψ : W ⊂ R3 → K around p the third component of the vector (Dψ)−1 (n(q)) is negative. The vector field n is called the outward unit normal field to the boundary ∂K. Since ∂K is an orientable two-dimensional manifold in R3 we can use the following characterization (see Spivak [59, Problem 5-14, p. 121]): There exists an open set U ⊂ R3 and a differentiable function f : U → R such that ∂K = f −1 (0) and ∇f (p) 6= 0 for all p ∈ ∂K. Moreover, n(p) = ±∇f (p)/ |∇f (p)|. Let K be a rigid body. We assume that the motion of K is described by a family of rigid body transformation {Φt }t∈[0,T ] , where T > 0. By a rigid body transformation we understand an orientation preserving isometry from R3 to R3 . Such transformations can always be represented as the composition of a translation and a rotation. A rotation is a linear transformation which can represented by an orthogonal matrix R with det R = 1. In view of Euler’s theorem (see Lax [39, p. 141]) every rotation matrix R 6= I has a uniquely defined axis of rotation equal to the eigenspace corresponding to the eigenvalue λ = 1. The set of all 3 × 3 rotation matrices form a group denoted by SO(3). Thus any rigid body transformation T can be represented in the form (5). T (p) = Rp + a. for all p ∈ R3 ,. where R ∈ SO(3) is a rotation matrix and a ∈ R3 is a translation vector. The transformation defined by (5) corresponds to first rotating the point p about the origin by R and then translating the point Rp by the vector a, or what is equivalent, first translating the point p by the vector a and then rotating the point p + a about the point a by R. Note also the T −1 is a rigid body transformation given by T −1 (p) = RT (p − a).. Given some point r ∈ R3 the rigid body transformations {Φt } are assumed to have the following representation: (6). Φt (p) = R(t)(p − r) + r + a(t),. p ∈ R3.

(14) 4. INTRODUCTION. where R : [0, T ] → SO(3) and a : [0, T ] → R3 are continuously differentiable functions such that (1) R(0) = I (2) and a(0) = 0. This implies that Φ0 is the identity transformation. The first two terms in the right side of (6) corresponds to a rotation with respect to the point r (center of rotation at time t = 0). The third term corresponds to translational motion. The infinitesimal generator of the rotational motion is by definition the matrix T ˙ A(t) = R(t)R(t) . By differentiating the relation R(t)R(t)T = I one deduces that A(t) is antisymmetric, i.e.   0 −ω3 (t) ω2 (t) 0 −ω1 (t) . A(t) =  ω3 (t) −ω2 (t) ω1 (t) 0 The vector. ω(t) = (ω1 (t), ω2 (t), ω3 (t)) lies in the nullspace of A(t) and is called the instantaneous angular velocity vector. The one-dimensional subspace spanned by ω(t) is called the instantaneous axis of rotation. The motion of a point with initial configuration p is given by the curve p(t) = Φt (p), t ∈ [0, T ]. The velocity of the moving point is obtained by differentiating the expression ˙ for p(t). Using the relation R(t) = A(t)R(t) one finds that ˙ ˙ ˙ p(t) = A(t)R(t)(p − r) + a(t) = A(t)(p(t) − Φt (r)) + a(t) which can be written by using the cross product as (7). ˙ ˙ p(t) = ω(t) × (p(t) − r(t)) + a(t). where r(t) = Φt (r) = r + a(t). Thus the rotational motion of a moving point p(t) is determined by ω(t) and the position of p(t) relative to the instantaneous center of rotation r(t). The evolution of K at time t is defined as Kt = {Φt (p) : p ∈ K}..

(15) 3. GEOMETRY AND KINEMATICS OF RIGID BODIES. 5. Since Φt : R3 → R3 is a diffeomorphism it is clear that Kt and ∂Kt are orientable manifolds of dimensions 3 and 2 respectively. The exterior unit normal field of ∂Kt is denoted by nt . One checks that nt (p) = R(t)n(Φ−1 t (p)) for all p ∈ ∂Kt . Lemma 3.1. Suppose that the rigid body K is contained in the ball . B(r, R) = p ∈ R3 : |p − r| < R. of center r ∈ R3 and of radius R > 0. Then for each t ∈ [0, T ],. (8). Kt ⊂ B(r(t), R).. Proof. Applying Φt to K ⊂ B(r, R). and using that balls are invariant with respect to rotations, we obtain (8). Lemma 3.2. Fix t ∈ [0, T ]. Let b = (b1 , b2 , b3 ) be a point on the boundary ∂Kt . Then (9). e3 · nt (b) 6= 0. if and only if there exist an open subset Ω of R2 containing (b1 , b2 ) and a differentiable function h : Ω → R, such that . b3 = h(b1 , b2 ) and (x, y, h(x, y)) : (x, y) ∈ Ω ⊂ ∂Kt .. Proof. In view of the characterization of orientable two-dimensional manifolds we have ∂Kt = ft−1 (0) with ft = f ◦ Φ−1 for some f : U ⊂ t R3 → R. In particular ft (p) = 0 for all p ∈ ∂Kt . Writing p = (x, y, z) and differentiating ft with respect to z, we obtain

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(17) ∂ft −1 T

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(19) (p) = ∇f (Φ−1 t (p)) · R(t) e3 = ± ∇f (Φt (p)) (e3 · nt (p)). ∂z Thus (9) is equivalent to ∂ft (10) (b) 6= 0. ∂z Suppose that (10) holds. By the implicit function theorem, there exist an open set Ω in R2 containing (b1 , b2 ) and an open set V in R containing b3 such that for each (x, y) ∈ Ω there exists a unique h(x, y) ∈ V with ft (x, y, h(x, y)) = 0, so (x, y, h(x, y)) ∈ ∂Kt . The function h is differentiable. The reverse implication is obvious. .

(20) 6. INTRODUCTION. 4. The domain Consider two rigid bodies with initial configuration K + and K − − whose motions are given by the transformations Φ+ t and Φt respectively. We assume that there exists positive numbers R+ and R− such that (1) r± = (0, 0, ±R± ) is an interior point of K ± , i.e. r± ∈ K ± −∂K ± (2) K ± ⊂ B(r± , R± ) (3) Φ± t can be represented in the form (6), i.e. ± ± ± ± Φ± t (p) = R (t)(p − r ) + r + a (t), ± with a± (t) = (a± 1 (t), a2 (t), 0) (4) the velocity field (or rigid flow) corresponding to Φ± t is given by. v ± (p, t) = ω ± (t)(p − r ± (t)) + a˙ ± (t),. ± where r ± (t) = Φ± t (r ).. Note that the plane Σ0 = {(x, y, z) ∈ R3 : z = 0}. is always tangent to the balls B(r ± (t), R± ). Hence Lemma 3.1 asserts − + + that the moving bodies Kt+ = Φ+ t (K) and Kt = Φt (K ) are always 0 separated by the plane Σ . This condition is essential in hydrodynamic lubrication. Our final assumption is that at each time t ∈ [0, T ] there exists an open set Ω(t) ⊂ R2 and a positive function h+ ( ; t) defined on Ω(t) satisfying the hypotheses of Lemma 3.2 for some point b+ ∈ ∂Kt+ and a negative function h− ( ; t) defined on Ω(t) satisfying the hypotheses of Lemma 3.2 for some point b− ∈ ∂Kt− such that the region in space defined by Ft = {(x, y, z) : (x, y) ∈ Ω(t) and h− (x, y; t) < z < h+ (x, y; t)} is occupied by fluid. The fluid film Ft is the time dependent domain for the lubrication problem. Note that the lubrication problem has moving boundaries. The upper boundary of the fluid film Ft is the surface . Σ+ (t) = (x, y, z) : (x, y) ∈ Ω(t) and z = h+ (x, y; t) ⊂ ∂Kt+ and the lower boundary of the film is the surface . Σ− (t) = (x, y, z) : (x, y) ∈ Ω(t) and z = h− (x, y; t) ⊂ ∂Kt− ..

(21) 4. THE DOMAIN. 7. Figure 1. The initial configuration of the rigid bodies K + and K − .. Figure 2. The fluid domain Ft .. Sometimes it is convenient that Ω does not depend on t. Then one must assume that there exists some open set in R2 denoted by Ω, such that Ω⊂. \. t∈[0,T ]. Ω(t)..

(22) 8. INTRODUCTION. 5. Boundary conditions Fluid particles adjacent to the boundary of a rigid body are assumed to move with the same velocity as the boundary. This is the so called no-slip condition. As seen above, the velocity of any point on ∂Kt± is determined by Φ± t . Thus we have the following boundary conditions for u: ( v + (p, t) = ω + (t) × (p − r + (t)) + a˙ + (t), p ∈ Σ+ u(p, t) = v − (p, t) = ω − (t) × (p − r − (t)) + a˙ − (t), p ∈ Σ− . These are boundary conditions of Dirichlet type.. Remark 5.1. Since rigid body transformations are volume preserving it holds that div v ± = 0. This is useful in computations. We know that the geometry of ∂K ± and the rigid body motion Φ± t determines the function h± . Clearly, this dependence imposes some constraints on the function h± . To establish this we consider a point + + p(t) = Φ+ t (p), where p ∈ ∂K , on the moving surface Σ . That is, we assume that p = (p1 , p2 , p3 ) satisfies p3 (t) = h+ (p1 (t), p2 (t); t) for all t in some open subinterval of [0, T ]. Differentiating this expression w.r.t. t, using the chain rule, gives p˙3 =. ∂h+ ∂h+ ∂h+ p˙1 + p˙2 + . ∂x ∂y ∂t. + Since p˙ is determined by Φ+ t according to (7), the function h must satisfy the following compatibility condition. ∂h+ (x, y; t), ∂t for each point p = (x, y, h+ (x, y; t)) on the boundary Σ+ . Since v + is assumed to be given, (11) is a quasilinear first order differential equation for h+ with initial condition at t = 0 determined by the geometry of K + , that is a transport equation. Moreover, assuming no-slip at the boundary, one sees that (11) is equivalent to (11). (12). v + (p, t) · (e3 − ∇h+ (x, y; t)) =. ∂h+ u · n+ = p ∂t 1 + |∇h+ |2. on Σ+. where n+ denotes the outward unit normal field to Σ+ , which is a boundary condition of Neumann type for u. Of course, similar conditions hold at the lower boundary Σ− ..

(23) 6. ASYMPTOTIC ANALYSIS. 9. 6. Asymptotic analysis We analyze the asymptotic behavior of the equations of motion of a compressible fluid in a thin film. The goal is to arrive at the thin-film equations from which the Reynolds equation is derived. To this end, for each 0 < ≤ 1 we shall consider the following lubrication problem. Let Ω be an open subset of R2 with smooth boundary. Define the film Ft = {(x, y, z) : (x, y) ∈ Ω, h− (x, y; t) < z < h+ (x, y; t)}, where h+ and h− are continuously differentiable functions defined on Ω × [0, T ] such that −1 ≤ h− < 0 < h+ ≤ 1. To ensure that the surfaces are separated we must also assume that there exists a positive constant α such that h+ − h− ≥ α. In this sense denotes the thickness of the film Ft . The motions of the rigid surfaces Σ,± = {(x, y, z) : (x, y) ∈ Ω, z = h± (x, y; t)} are assumed to be described by rigid body transformations Φ,± defined t ± ± ± ± as follows. Given a constant vectors ω = (ω1 , ω2 , ω3 ), constants R± > ± 0 and continuously differentiable functions a± 1 and a2 defined on [0, T ] set     0 −ω3± ω2± 0 r,± =  0  A,± =  ω3± 0 −ω1±  ±R± / 0 −ω2± ω1±  ±  a1 (t)  a± (t) = a± 2 (t) . 0 For q ∈ R3 , we define. ,± Φ,± )(q − r,± ) + r,± + a± (t). t (q) = exp(tA. In other words, Φ,± describes a rotation with respect to the point r,± t with constant angular velocity vector ω ,± = (ω1± , ω2± , ω3± ) and a translation by the vector a± (t). One can show that the velocity v ,± of the surface Σ,± admits an expansion in the form (13). v ,± = v 0,± + v 1,± + 2 v 2,± ..

(24) 10. INTRODUCTION. Indeed, for any (x, y) ∈ Ω  ±    ±  ω1 x − a± a˙ 1 (t) 1 (t) ± ,± ±      v (x, y, t) = ω2 × + a˙ ± y − a2 (t) 2 (t) ± ± ± h (x, y; t) ∓ R / 0 ω3  ±  ± ± ± ± −ω3 (y − a2 (t)) ∓ ω2 R + a˙ 1 (t) ±   = ω3± (x − a± ˙± 1 (t)) ± ω1 R + a 2 (t) 0 {z } | v 0,±. .  0  0 +  ± ± ± ± ω (y − a2 (t)) − ω2 (x − a1 (t)) | 1 {z }. (14). +. v 1,±  ± ± ω2 h (x, y; t) 2 −ω1± h± (x, y; t) ..  |. 0 {z. v 2,±. }. is Remark 6.1. The lubrication problem defined by Ft and Φ,± t somewhat artificial because for it to have any significance to the real world, Σ,± must be a portion of the boundary of some rigid body whose motion is given by Φ,± t . To make the problem realistic, it must be required that for each the function h± satisfies the compatibility condition given by (11), i.e. ∂h± (x, y; t) in Ω. ∂t and equating terms of equal powers. v ,± (x, y, t) · (e3 − ∇h± (x, y; t)) = Inserting the expansion (13) for v ,± of we obtain (15). 0 :. v 0,± · e3 = 0. (16). 1 :. v 1,± · e3 = v 0,± · ∇h± +. (17). 2 :. (18). 3 :. v 2,± · e3 = v 1,± · ∇h±. 0 = v 2,± · ∇h± .. ∂h± ∂t. From (14) we see that (15) and (17) holds by the definition of Φ,± t . It is unlikely however that both (16) and (18) can be satisified for the same function h± . We conclude therefore that the compatibility condition for this problem is given by (16) alone. If is small this is a good.

(25) 6. ASYMPTOTIC ANALYSIS. 11. approximation, namely the error is of the order 2 , of realistic rigid body motion. Consider the Navier–Stokes system (19) (20). ρ ∂t u + ρ (u · ∇)u = −∇p + (λ + µ)∇(div u ) + µ∆u ∂t ρ + div(ρ u ) = 0. in the domain Ft , t ∈ [0, T ]. The boundary condtitions for u arising from the no-slip assumption, at the upper and lower boundaries are u |Σ,+ = v ,+. (21). and u |Σ,+ = v ,+ .. Moreover we assume that the following density-pressure relation holds ρ (p ) = ρa exp(2 βp ),. (22). where ρa > 0 and β are constants characteristic to the fluid considered. We want to study the asymptotic behavior of the solutions to this boundary value problem as tends to zero. Heuristical arguments (such as a dimensional analysis of the problem) leads us to seek a solution of the form (23). u =. ∞ X. k uk (x, y, z/, t),. k=0. p =. ∞ X. k−2 pk (x, y, z/, t). k=0. where uk = (uk , v k , wk ), k = 0, 1, 2, . . . . Inserting the above expansion for p into (22) we obtain the following expansion for ρ : (24) ρ (p ) = ρa exp(βp0 ) 1 + βp1 + 2 βp2 + (βp1 )2 + · · · . | {z } ρ0. Remark P 6.2. kActually, it suffices to postulate an expansion of the form ρ = ∞ k=0 ρk for our method to work.. The idea of asymptotic expansion is to plug the expansions (23) and (24) into (19)–(21) and equate like powers of . By changing variables from z to z we shall consider the resulting equations in the constant (w.r.t. ) domain Ft = Ft1 with upper boundary Σ+ and lower boundary Σ− . For simplicity we employ here the abbreviations ∂t = ∂/∂t, ∂x = ∂/∂x, . . . etc. for partial derivatives. Writing out only the non-trivial equations thus obtained, the equations of motion (19) are broken down.

(26) 12. INTRODUCTION. into (25). −3 :. (26). −2 :. (27). −2 :. (28). −2 :. (29). −1 :. (30). −1 :. (31). −1 :. 0 = −∂z p0. 0 = −∂x p0 + µ∂z2 u0 0 = −∂y p0 + µ∂z2 v 0. 0 = −∂z p1 + (λ + µ)∂z2 w0 + µ∂z2 w0. ρ0 w0 ∂z u0 = −∂x p1 + (λ + µ)∂x ∂z w0 + µ∂z2 u1 ρ0 w0 ∂z v 0 = −∂y p1 + (λ + µ)∂y ∂z w0 + µ∂z2 v 1. ρ0 w0 ∂z w0 = −∂z p2. + (λ + µ)(∂z ∂x u0 + ∂z ∂y v 0 + ∂z2 w1 ) + µ∂z2 w1. (32). and so on for 0 , 1 , . . . . Repeating this procedure for the boundary condition (21) yields (33). 0 :. (34). 1 :. (35). 2 :. u0 |Σ+ = v 0,+. u1 |Σ+ = v 1,+. u2 |Σ+ = v 2,+. u0 |Σ− = v 0,−. u1 |Σ− = v 1,−. u2 |Σ− = v 2,− .. For the continuity equation (20) we obtain (36). −1 :. (37). 0 :. ∂z (ρ0 w0 ) = 0 ∂t ρ0 + ∂x (ρ0 u0 ) + ∂y (ρ0 v 0 ) + ∂z ρ0 (w1 + βp1 w0 ) = 0. and so on. Let us analyze the equations obtained so far. Equation (25) says that p0 , does not depend on z, i.e. p0 = p0 (x, y, t). From (36) we then conclude that w0 is constant along lines parallel to the vector e3 and from (33) we see that w0 = 0 on Σ+ ∪ Σ− . Therefore we conclude that w0 = 0 in Ft . Summing up, we have arrived at the so called thin film equations: (38) (39). 1 0 ∇p µ ∂t ρ0 + ∂x (ρ0 u0 ) + ∂y (ρ0 v 0 ) + ρ0 ∂z w1 = 0 ∂z2 u0 =. where p0 does not depend on z and u0 = (u0 , v 0 , 0) satisfies the boundary conditions (33) and w1 = e3 · u1 satisfies the boundary conditions (34)..

(27) 7. THE REYNOLDS EQUATION. 13. 7. The Reynolds equation Since yield. p0. does not depend on z we can integrate the equation (38) to. z2 ∇p0 + za + b, 2µ where a and b are functions of x, y and t that can be determined from the boundary conditions. It is readily checked that u0 =. (40). u0 =. h+ − z 0,− (z − h+ )(z − h− ) 0 z − h− 0,+ v + + v ∇p + + − 2µ h −h h − h−. satisfies the boundary conditions (33). Next, let ϕ = ϕ(x, y) be a smooth function with compact support in Ω. Multiply (39) by ϕ and integrate over Ft . By the Gauss–Green theorem we obtain Z 0= ϕ ∂t ρ0 + ∂x (ρ0 u0 ) + ∂y (ρ0 v 0 ) + ρ0 ∂z w1 dx dy dz ZFt = ϕ∂t ρ0 − ∂x ϕ ρ0 u0 − ∂y ϕ ρ0 v 0 dx dy dz (41) Ft Z ϕρ0 (u0 , v 0 , w1 ) · n dS + ∂Ft. = I1 + I2 .. By Fubini’s theorem the volume integral I1 can be written as Z Z h+ 0 0 0 0 0 I1 = ϕ∂t ρ − ∂x ϕ ρ u − ∂y ϕ ρ v dz dx dy Ω. =. Z. Ω. h−. +. −. 0. 0. ϕ(h − h )∂t ρ − ∇ϕ · ρ. Z. h+. h−. 0. . u dz dx dy.. From (40) we calculate Z h+ (h+ − h− )3 0 h+ − h− 0,+ u0 dz = − ∇p + (v + v 0,− ) 12µ 2 − h h3 h =− ∇p0 + (v 0,+ + v 0,− ), 12µ 2 where h = h+ − h− . By the two-dimensional Gauss–Green theorem we obtain 3 0 Z h ρ hρ0 0,+ 0 0 0,− (42) I1 = ϕh∂t ρ + ϕ div − ∇p + (v + v ) dx dy 12µ 2 Ω.

(28) 14. INTRODUCTION. The boundary of Ft can be written ∂Ft = Σ+ ∪ Σ− ∪ Γ. The outward unit normal to Σ± is e3 − ∇h± . n± = ± q 2 ± 1 + |∇h |. Thus the compatibility condition (16) can be written as the Neumann condition: ∂h± (u0 , v 0 , w1 ) · n± = ± q ∂t on Σ± 2 1 + |∇h± | for the vector field (u0 , v 0 , w1 ). Since ϕ vanishes on Γ we can calculate the surface integral Z I2 = ϕρ0 (u0 , v 0 , w1 ) · n dS Σ+ ∪Σ−. (43). ∂h+ ∂h− Z − ∂t = ϕρ0 q ∂t dS + ϕρ0 q dS + − 2 Σ Σ 1 + |∇h+ | 1 + |∇h− |2 Z Z + ∂h− 0 ∂h ϕρ = dx dy − ϕρ0 dx dy ∂t ∂t Ω ZΩ ∂h ϕρ0 = dx dy. ∂t Ω Z. Inserting (42) and (43) into (41) we obtain 3 0 Z ∂ h ρ hρ0 0,+ ϕ (hρ0 ) + ϕ div − ∇p0 + (v + v 0,− ) dx dy = 0. 12µ 2 Ω ∂t. Since this holds for all ϕ ∈ Cc∞ (Ω) we conclude that p0 satisfies the Reynolds equation (44) 3 0 ∂ h ρ hρ0 0,+ 0 0 0,− (hρ ) + div − ∇p + (v + v ) = 0 in Ω × [0, T ]. ∂t 12µ 2 8. Forces on the bounding surfaces We continue the asymptotic analysis by calculating the stresses on the rigid boundaries. To this end we consider the surfaces Σ,+ and Σ,− and denote by f ,+ and f ,− the corresponding surface forces. The constitutive relation is (45). T = (−p + λ trace D )I + 2µD ,.

(29) 8. FORCES ON THE BOUNDING SURFACES. 15. where D = (∇u + (∇u )T )/2. Thus Z Z T |z=h± ±(∇h± − e3 ) dx dy. T n,± dS = (46) f ,± = Σ,±. Ω. We have the formal expansions (recall that w0 = 0)   0 0 ∂z u0 1  0 0 ∂z v 0  D = 2 ∂ u0 ∂ v 0 0 z z   0 2∂x u ∂y u0 + ∂x v 0 ∂z u1 1 2∂y v 0 ∂z v 1  + O() + ∂x v 0 + ∂y u0 2 1 1 ∂z u ∂z v 2∂z w1 = −1 D 0 + D 1 + O(). trace D = ∂x u0 + ∂y v 0 + ∂z w1 + O() T = −2 (−p0 I) + −1 −p1 I + 2µD 0. . + 0 −p2 I + λ(∂x u0 + ∂y v 0 + ∂z w1 )I + 2µD 1 + O(). = −2 T 0 + −1 T 1 + 0 T 2 + O().. Inserting this into (46) gives Z f ,± = ± −2 (−T 0 e3 ) + −1 (T 0 ∇h± − T 1 e3 ) Ω. (47). + 0 (T 1 ∇h± − T 2 e3 ) + O() dx dy. = −2 f 0,± + −1 f 1,± + 0 f 2,± + O(), where f 0,± f 1,±.   0 = ± −T 0 e3 dx dy = ±  0  dx dy Ω Ω p0 Z = ± T 0 ∇h± − T 1 e3 dx dy Ω      

(30) Z 0 ∂x h± ∂z u0

(31)

(32) = ± −p0 ∂y h±  +  0  − µ ∂z v 0 

(33)

(34) dx dy Ω

(35) 0 p1 0 ± z=h   Z 0 1 0,+ h 0 0,− 0 ±   − v ) dx dy. = ± −p ∇h + 0 − µ ± ∇p + (v 2µ h Ω p1 Z. Z. People who work in the field of hydrodynamic lubrication distinguish between “load carrying force” which is defined as the contribution from.

(36) 16. INTRODUCTION. normal stresses and “hydrodynamic friction force” which is due to shear stresses. In view of the expansion (47) this distinction is also quite (but not completely) natural in view of the separation in terms of order of . We therefore define   Z 0 0,±  0  dx dy f load = ± Ω p0   Z 0 1,±  0  − p0 ∇h± dx dy f load = ± Ω p1 Z µ h f 1,± − ∇p0 ∓ (v 0,+ − v 0,− ) dx dy. friction = 2 h Ω. Then (48). 1,± 1,± −1 f ,± = −2 f 0,± load + (f load + f friction ) + O(1).. 1,± Remark 8.1. Note that f 0,± load is parallel to e3 whereas f friction is ,± perpendicular to e3 . Let f ,± that is ⊥e3 denote the component of f perpendicular to e3 . Then we have Z ,± 1,± 0 ± −1 p ∇h dx + O(1). f ⊥e3 = f friction ∓ Ω. One could thus argue that the quantity Z h µ 1,± f ⊥e3 = − ∇p0 ∓ (v 0,+ − v 0,− ) ∓ p0 ∇h± dx dy 2 h Ω. − would be more interesting to compute than f 1,± friction . If, say h = 0, then 1,− 1,− clearly the expressions for f ⊥e3 and f friction coincide.. 9. Solution of the lubrication problem We summarize the result of the asymptotic analysis carried out above. Theorem 9.1 (Fundamental theorem of lubrication theory). Given a domain Ω in R2 and functions h+ and h− defined on Ω × [0, T ], we consider a space-time domain F , defined by h± and Ω through . F = (x, y, z, t) : (x, y) ∈ Ω, t ∈ [0, T ], h+ (x, y; t) < z < h− (x, y; t) .. We assume that the velocity fields of the moving surfaces Σ+ : z = h+ (x, y; t) and Σ− : z = h− (x, y; t) are given and denoted them by (v1+ , v2+ , v3+ ).

(37) 9. SOLUTION OF THE LUBRICATION PROBLEM. 17. and (v1− , v2− , v3− ) respectively. Set v + = (v1+ , v2+ ),. v − = (v1− , v2− ).. We assume that Σ+ and Σ− are rigid surfaces, which implies that h+ and h− satisfies ∂h± + v ± · ∇h± = v3± ∂t in Ω × [0, T ]. Then the velocity field u of a viscous fluid with viscosity µ occupying F is given by (49). (50). u=. z − h− + h+ − z − (z − h+ )(z − h− ) ∇p + v + v + O(h), 2µ h h. where h = h+ − h− and p and ρ satisfy the Reynolds equation ∂ hρ + h3 ρ − (51) (hρ) + div − ∇p + (v + v ) = 0 in Ω × [0, T ]. ∂t 12µ 2. The error in the approximation (50) is thus small provided that the film is thin, i.e. h L where L is the characteristic length of Ω, and that the surfaces move in almost parallel directions, i.e. v3+ and v3− are both O(h). Moreover, the force exterted by the fluid on the surface Σ± is given by   0 −1 f± = ±  0  + f± friction + O(h ) fload where Z (52) fload = p dx dy Ω Z h µ ± (53) f friction = − ∇p ∓ (v + − v − ) dx dy 2 h Ω

(38) ±

(39) −2

(40)

(41) and fload is O(h ) whereas f friction is O(h−1 ) (see also Remark 8.1). Remark 9.2. Since ρ is assumed to be a function of p, the Reynolds equation (51) is nonlinear.. Remark 9.3. Equation (51) is identical to equation (7.58) in Hamrock [33, p. 158] but the derivation presented here is different. By using the compatibility condition (49) for the functions h+ and h− one can express (51) in various equivalent ways (see e.g. Hamrock [33, p. 151]). Although (51) is perhaps the most elegant way to write the Reynolds equation, it may not be the most practical. This motivates why (49) (expressing constrained rigid body motion) deserves to be stated as separate equations for h+ and h− ..

(42) 18. INTRODUCTION. 9.1. Simplified models. If we assume that ρ is constant in (51), we obtain the incompressible Reynolds equation ∂h h3 h + − (54) + div − ∇p + (v + v ) = 0 ∂t 12µ 2 Using the compatibility condition (49) and that div v ± = 0 (by assumption), we find that div h(v + + v − ) + 2∂t h = ∇(h+ − h− ) · (v + + v − ) + 2(v3+ − ∇h+ · v + − v3− + ∇h− · v − )). = ∇(h+ + h− ) · (v − − v + ) + 2(v3+ − v3− ). Thus, the incompressible Reynolds equation can also be written in the form (55). div(h3 ∇p) = 6µ∇(h+ + h− ) · (v − − v + ) + 12µ(v3+ − v3− ).. Observe that in this form, only the relative velocity enters the equation. In particular, if h− = 0 and v3+ = v3− then h = h+ and (55) reduces to (56). div(h3 ∇p) = 6µ∇h · (v − − v + ).. In this case the friction force on the lower surface Σ− becomes Z h µ − − ∇p + (v + − v − ) dx dy. (57) f friction = 2 h Ω Remark 9.4. Equation (56) is more or less the form in which it was first found by its discoverer Osborne Reynolds (1842–1912). For more details we refer to Reynolds’ paper [57] from 1886 (that is, exactly 125 years ago). According to Hamrock [33, p. 147] the compressible Reynolds equation (51) is due to Harris (1913). If we assume that ρ depends on p according to ρ = ρa exp(βp), we can write (51) as a linear equation for ρ: ∂ hρ + h3 − (58) (hρ) + div − ∇ρ + (v + v ) = 0. ∂t 12βµ 2 9.2. Rigorous justification. The method of asymptotic expansion, which lead to Theorem 9.1 is only formal and by mathematical standards not rigorous. To prove the theorem one needs error estimates,.

(43) 10. HISTORICAL REMARKS. 19. in some suitable norm, for the Navier–Stokes system (19)–(20) to deduce that at least. lim 2 p − p0 = 0 →0. (z/ − h+ )(z/ − h− ) 0. lim u − ∇p →0 2µ z/ − h− 0,+ h+ − z/ 0,− . − v − v = 0, h h 0 where p denotes the solution of equation (44). Justifying the Reynolds equation in its general form (51) is a rather delicate task as it involves a detailed analysis of the compressible Navier–Stokes system. However, the Reynolds equation has been justified rigorously, starting from intermediate models. For example, the reduction of the incompressible Stokes equations to the Reynolds equation was proved by Bayada and Chambat [9] in 1986. In 2007, Nazarov and Videman [50] proved error estimates for the stationary incompressible Navier–Stokes system in curvilinear coordinates including also higher order inertial terms. An overview of work in this directions is given in the introduction of [69]. The compressible case was considered in 2010 by Maruˇsić-Paloka and Starˇcević [45], who justified the Reynolds equation for gas lubrication assuming a linear relationship between pressure and density. 10. Historical remarks Reynolds [57] applied his equation to a cylindrical journal revolving in a cylindrical bearing (see Figure 3), assuming that the surfaces are “nearly parallel” and that the radii of curvature of both bearing and journal are large compared with the thickness of the film. Reynolds’ work provided a theoretical basis for the understanding and design of bearings and proved to be in close agreement with a series of careful experiments that had been conducted by Tower. Although primarily known as a prominent figure in fluid dynamics, Reynolds also contributed to other areas of tribology through his fundamental work on rolling friction. For a long time, the applicability of Reynolds’ equation was limited to obtaining two-dimensional analytical solutions. Closed-form solutions were only known for infinitely long bearings (Sommerfeld 1904) and infinitely short bearings (Michell 1905 and Ocvirk 1952). But with the advent of fast computers, the Reynolds equation has become an indispensable tool in bearing design. Compared to more general models describing fluid flow such as the Navier–Stokes equations, the Reynolds equation is linear, two-dimensional and gives the asymptotically correct solution as.

(44) 20. INTRODUCTION. Figure 3. A schematic description of the journal bearing in Tower’s experimental set up (after Reynolds [57]). The bearing was immersed in an oil bath. the film thickness h tends to zero. Experience shows that the Reynolds equation is a reasonably good approximation for many problems in hydrodynamic lubrication. The transition between the three-dimensional models for viscous flows 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 not only in fluid film bearings but also in e.g. gas pipelines, capillaries, oceans and the atmosphere. As shown by many rigorous studies, in the mathematical literature [9, 12, 24, 25, 28, 44, 48, 49] and in the engineering literature [8, 20, 29, 62, 67], the scaled Reynolds equation gives an O(h) approximation, i.e. a zeroth-order approximation, of the pressure distribution. For lubrication with high Reynolds number, e.g. lubrication of a rapidly rotating shaft, Reynolds’ approximation becomes rather crude. The need for higher-order approximations of pressure and velocity fields in hydrodynamic lubrication has been confirmed by numerous theoretical and experimental studies, see the references cited in [43, 50]. Recent investigations [43, 47, 50] in this direction have lead to modified nonlinear Reynolds-type equations, containing terms that arise from inertial and curvature effects. Hydrodynamic lubrication with non-Newtonian fluids is also an active research field. A nonlinear Reynolds-type equation accounting for non-Newtonian effects has been proposed in [34]. 11. Surface roughness and homogenization As shown above, the Reynolds equation can take a variety of forms, see also [29, 60]. We consider here a very simple case. If the lower surface is a portion of the plane x3 = 0 and the upper surface is the.

(45) 11. SURFACE ROUGHNESS AND HOMOGENIZATION. 21. corresponding portion of the surface x3 = h(x1 , x2 ), where h is a smooth function bounded from below by some positive constant, then a simple form of Reynolds equation reads (59). ∂h div h3 ∇p = 6µv ∂x1. The unknown p = p(x1 , x2 ) is the pressure distribution and h is commonly referred to as the thickness of the film. The lower surface is assumed to move at constant speed v in the x1 -direction, whereas the other surface remains fixed. A no slip condition is prescribed at the bounding surfaces. The lubricant viscosity µ is taken as constant. Equation (59) is assumed to hold in an open subset Ω of R2 and the boundary condition is p = 0 on ∂Ω. Even when the error in the thin-film approximation is tolerably small, at least two other effects have been recognized that could render the Reynolds equation invalid. First, there is the effect of molecular slip at the boundary, causing the macroscopic velocity of the fluid near the boundary to deviate from that of the adjacent surface. Such effects become apparent in magnetic storage devices consisting of a mechanical head flying over a rotating disk, the lubricant being air. Modifications to the Reynolds equation so as to compensate for this effect have been suggested by Burgdorfer. Second, there is the effect of surface roughness. Technical surfaces are never perfectly smooth due to imperfections in the manufacturing process. Almost smooth surfaces can only be manufactured at extremely high costs. Roughness may cause increased wear and is therefore usually considered undesirable in mechanical contacts. However, in tilted slider bearings and journal-type bearings, it has been observed that the hydrodynamic performance can be improved by adding roughness. Hence, artificially machined roughness (or texture) can also be regarded as a design parameter. In macroscopic fluid models the surface micro-topography is usually neglected for laminar flow, but when the lubricant film is sufficiently thin even small roughness becomes significant. According to Elrod [30], the first theoretical studies of surface roughness appeared in the 1950s. For a review of the state of the art the reader may consult Elrod’s paper [31], the monograph [68, Ch. 7] and the doctoral thesis [4]. Surface roughness enters the Reynolds equation through the function describing the film thickness. The usual ansatz in statistical treatments is h = g + R where g is a function representing the “global” film thickness and R is a stochastic variable representing the roughness (in deterministic treatments R corresponds to a specific surface description). Since.

(46) 22. INTRODUCTION. Figure 4. A periodically rough surface.. the roughness is random, the solution of the Reynolds equation must be averaged at some point in the calculations. For this various techniques have been suggested by many authors, e.g. Tzeng and Saibel, Christensen, Elrod, Chow and Saibel, Patir and Cheng and Phan-Thien [22, 23, 30, 32, 53, 56, 64]. There has been some controversy as to the validity of the Reynolds equation for rough surfaces. Elrod [30] and Sun and Chen [61] think that the Reynolds equation is inadequate when the roughness wavelength ε becomes smaller than or comparable in magnitude with the film thickness h, arguing that the basic assumptions used to derive the Reynolds equation no longer hold. In this case the Stokes equations can be used instead. Elrod proposed to classify roughness into two categories: “Reynolds roughness” and “Stokes roughness”. Reynolds roughness corresponds to the case h ε. Early studies [55, 61] showed that the Reynolds equation and Stokes equations may lead to conflicting results, but an explanation to this is difficult because of the various assumptions introduced by the authors. The first rigorous attempt to clarify the concepts of Reynolds and Stokes roughness is that by Bayada and Chambat [9] (see also [10, 46]). Using homogenization theory they study the Stokes equations in a thin domain bounded by a periodically rough surface and a smooth plane when ε and h tend to zero. Depending on the value of the parameter λ = h/ε different equations are obtained in the limit. Three situations apply:.

(47) 11. SURFACE ROUGHNESS AND HOMOGENIZATION. 23. (1) λ → constant > 0. A two-dimensional equation is obtained with coefficients that depend on the surface micro-topography and λ. (2) λ → 0 (Reynolds roughness). The homogenized Reynolds equation is obtained, i.e. first h → 0, then ε → 0. In agreement with previous studies. (3) λ → ∞ (Stokes roughness). A simple Reynolds equation is obtained with an effective film thickness. The same result is obtained if first ε → 0, then h → 0.. A remarkable conclusion of this study is that there is really no need to consider the Stokes roughness. This message does not quite seem to have assimilated into the engineering community, where considerable attention is given to the Stokes roughness, see the recent studies [40, 65]. We can only speculate why this is so, but perhaps the hypotheses about the roughness in [11] is considered to be too weak. This also suggests that more rigorous studies of thin film lubrication with two rough surfaces are needed. In the mathematical community, homogenization has become the dominant approach to studying the influence of surface roughness in lubrication. Homogenization theory is a set of mathematical techniques that are aimed at studying differential operators with rapidly oscillating coefficients, equations in perforated domains or equations subject to rapidly alternating boundary conditions. The first proof of a homogenization theorem was obtained by De Giorgi and Spagnolo around 1970. There exists today a vast literature on homogenization theory, see e.g. the books [16, 21, 26, 37, 52, 54, 63]. Let us qualitatively describe homogenization of equation (59) and surface roughness. The roughness enters the Reynolds equation through the film thickness h. It is assumed that the roughness is periodic with period ε. To this end let R be a function that is 1-periodic in both arguments and has mean value zero on the corresponding cell of periodicity. The film thickness entering the Reynolds equation is given by x hε (x) = g(x) + R ε resulting in the equation ∂ ∂pε ∂ ∂pε ∂hε (60) h3ε + h3ε = 6µv . ∂x1 ∂x1 ∂x2 ∂x2 ∂x1. Homogenization of (60) has been studied by Wall [66] who showed that the sequence of solutions pε converges to a limit p which is the solution.

(48) 24. INTRODUCTION. of an “averaged” or homogenized equation, 2 2 X X ∂p ∂g ∂bi ∂ aij = 6µv − . (61) ∂xi ∂xj ∂x1 ∂xi i=1. i,j=1. The averaged or homogenized coefficients aij and bi (1 ≤ i, j ≤ 2) are found by first solving v0 = v0 (x, y), v1 = v1 (x, y) and v2 = v2 (x, y) from the three periodic problems ∂ ∂ ∂h 3 ∂v0 3 ∂v0 (62a) h + h = 6µv ∂y1 ∂y1 ∂y2 ∂y2 ∂y1 ∂ ∂v1 ∂ ∂v1 ∂h3 (62b) h3 + h3 =− ∂y1 ∂y1 ∂y2 ∂y2 ∂y1 ∂ ∂v2 ∂ ∂v2 ∂h3 (62c) h3 + h3 =− ∂y1 ∂y1 ∂y2 ∂y2 ∂y2. in , where h(x, y) = g(x) + R(y) and denotes the cell of periodicity (usually the square (0, 1) × (0, 1)). Then use the averaging formulae ! Z ∂v1 ∂v2 ∂v0 a11 a12 b1 3 1 + ∂y1 ∂y1 ∂y1 (63) = h dy ∂v0 ∂v1 ∂v2 a21 a22 b2 1 + ∂y ∂y ∂y 2. 2. 2. to compute the coefficients. This gives the following homogenization algorithm: (1) Solve the local problems (62). (2) Compute the homogenized coefficients (63). (3) Solve the homogenized Reynolds equation (61). Some mathematically delicate questions that arise are in what sense pε converges to p as ε → 0 and whether the coefficients (63) are sufficently “nice” for (61) to be well posed. Heuristically, one can find the homogenized equation (61) and the local problems (62) by making the ansatz x x x + εp1 x, + ε2 p2 x, + ··· pε (x) = p x, ε ε ε. and plug this into (60). Equating terms of equal powers of ε and solving the obtained system of equations eventually yields the homogenization result. This is the formal method of asymptotic expansion described above. The homogenized Reynolds equation (61) contains no oscillating coefficents. Nevertheless it retains local information of the film thickness, thereby capturing the effects of roughness. It is in general different from.

(49) 11. SURFACE ROUGHNESS AND HOMOGENIZATION. 25. the Reynolds equation corresponding to mean film thickness, i.e. ∂ ∂ ∂g 3 ∂p 3 ∂p g + g = 6µv . ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 For general two-dimensional roughness the coefficients of the homogenized equation can 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 can be found. The first publication ever to use the word ‘homogenization’ in the context of hydrodynamic lubrication is probably [56]. However, already in 1973, Elrod [30] had proposed a very 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 homogenization by the “asymptotic expansion method” of Bakhvalov and Lions. Since Elrod’s pioneering work, ideas from homogenization theory have been frequently applied in hydrodynamic lubrication. One of the first rigorous studies in the field is that of Bayada and Chambat [11]. Subsequent studies pertain mostly to Reynolds roughness in various lubrication regimes, see e.g. the works [6, 13, 14, 15, 17, 27, 35, 41, 66]. For homogenization to be useful to e.g. bearing designers, an efficient numerical treatment of the homogenized equations is needed. Such aspects have been discussed in [18, 19]. Comparisons between homogenization and traditional averaging of Reynolds equation have been undertaken in [36, 38]. We summarize their findings: • A deterministic approach is limited by the resolution of the numerical method employed. A more cost-effective method is to compute the solution of the homogenized equation even though some local problems must be solved first. Actually the finer the roughness, the better the agreement between the homogenized and the deterministic solution. • Homogenization works regardless of the type of roughness. It is particularly well suited for anisotropic roughness, retaining information regarding the amplitude and the direction of the roughness. Some stochastic methods that have been proposed prove defective in this case. • Unlike some averaging methods that may lead to ambiguous results, the homogenized equation is uniquely determined. • Being a mathematical theory, homogenization is a completely rigorous approach, whereas many other methods are based on heuristics..

(50) 26. INTRODUCTION. (a) Deterministic pressure distribution (b) Homogenized pressure distribution. Figure 5. A comparison of deterministic and homogenized pressure distributions in a fluid film bearing (from Almqvist et al. [5]). 12. Outline of this thesis The papers I–V included in this thesis are all related to homogenization and hydrodynamic lubrication. I–III focus on theoretical aspects, whereas IV and V are more oriented towards specific applications. I is devoted to the homogenization of a Reynolds equation describing compressible flow. It can be noted that this equation does not fall under the standard treatment of linear parabolic equations, but requires some non-standard tricks. The main result of II is a multiscale (or reiterated) homogenization result for a nonlinear Reynolds-type equation. To this end we first develop some aspects of the theory of multiscale convergence introduced by Nguetseng, Allaire, Allaire and Briane and Lukkassen et al. [51, 2, 3, 42]. The proof of the main theorem relies on some wellknown results concerning decomposition of vector fields due to Weyl. The Weyl decomposition is further investigated in III. Reiterated homogenization makes it possible to analyze surface roughness with several characteristic wavelengths. How this is done in practice is explained in IV, where it is also shown how asymptotic expansion can be used to find the homogenized equation. To compare the homogenized solution with the solution of the deterministic (unaveraged) Reynolds equation, some numerical examples are also included. V is devoted to homogenization of a variational principle generalizing the unstationary incompressible Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced “variational bounds” (see [6]) are derived from a variational principle..

(51) 12. OUTLINE OF THIS THESIS. 27. Bounds can be seen as an economic alternative to computing the relatively costly homogenized solution and give a mathematical explanation to some heuristically proposed averaging techniques. For simple kinds of roughness the bounds even coincide with the homogenized solution. Several numerical examples are included to illustrate the utility of bounds..

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(55) BIBLIOGRAPHY. 31. 34. J.-H. He. Variational principle for non-Newtonian lubrication: Rabinowitsch fluid model. Appl. Math. Comput., 157(1):281–286, 2004. 35. M. Jai. Homogenization and two-scale convergence of the compressible Reynolds lubrication equation modelling the flying characteristics of a rough magnetic head over a rough rigid-disk surface. RAIRO Modél. Math. Anal. Numér., 29(2):199– 233, 1995. 36. M. Jai and B. Bou-Sa¨ıd. A comparison of homogenization and averaging techniques for the treatment of roughness in slip-flow-modified reynolds equation. Journal of Tribology, 124(2):327–335, 2002. 37. V. V. Jikov, S. M. Kozlov, and O. A. Ole˘ınik. Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif0 yan]. 38. M. Kane and B. Bou-Said. Comparison of homogenization and direct techniques for the treatment of roughness in incompressible lubrication. Journal of Tribology, 126(4):733–737, 2004. Cited By (since 1996): 10. 39. P. D. Lax. Linear algebra. Pure and Applied Mathematics (New York). John Wiley & Sons Inc., New York, 1997. A Wiley-Interscience Publication. 40. J. Li and H. Chen. Evaluation on applicability of reynolds equation for squared transverse roughness compared to cfd. Journal of Tribology, 129(4):963–967, 2007. 41. D. Lukkassen, A. Meidell, and P. Wall. Homogenization of some variational problems connected to the theory of lubrication. Internat. J. Engrg. Sci., 47(1):153– 162, 2009. 42. D. Lukkassen, G. Nguetseng, and P. Wall. Two-scale convergence. Int. J. Pure Appl. Math., 2(1):35–86, 2002. 43. S. Maruˇsić. Nonlinear Reynolds equation for lubrication of a rapidly rotating shaft. Appl. Anal., 75(3-4):379–401, 2000. 44. S. Maruˇsić and E. Maruˇsić-Paloka. Two-scale convergence for thin domains and its applications to some lower-dimensional models in fluid mechanics. Asymptot. Anal., 23(1):23–57, 2000. 45. E. Maruˇsić-Paloka and M. Starˇcević. Derivation of Reynolds equation for gas lubrication via asymptotic analysis of the compressible Navier-Stokes system. Nonlinear Anal. Real World Appl., 11(6):4565–4571, 2010. 46. A. Mikelić. Remark on the result on homogenization in hydrodynamical lubrication by G. Bayada and M. Chambat. RAIRO Modél. Math. Anal. Numér., 25(3):363–370, 1991. 47. C. M. Myllerup and B. J. Hamrock. Perturbation approach to hydrodynamic lubrication theory. Journal of Tribology, 116(1):110–118, 1994. 48. S. A. Nazarov. Asymptotic solution of the Navier-Stokes problem on the flow of a thin layer of fluid. Sibirsk. Mat. Zh., 31(2):131–144, 1990. 49. S. A. Nazarov and K. I. Piletskas. The Reynolds flow of a fluid in a thin threedimensional channel. Lit. Mat. Sb., 30(4):772–783, 1990. 50. S. A. Nazarov and J. H. Videman. A modified nonlinear Reynolds equation for thin viscous flows in lubrication. Asymptot. Anal., 52(1-2):1–36, 2007. 51. G. Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 20(3):608–623, 1989. 52. A. Pankov. G-convergence and homogenization of nonlinear partial differential operators, volume 422 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht, 1997..

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(59) HOMOGENIZATION OF A REYNOLDS EQUATION DESCRIBING COMPRESSIBLE FLOW ANDREAS ALMQVIST, JOHN FABRICIUS, AND PETER WALL Abstract. We homogenize a Reynolds equation with rapidly oscillating film thickness function hε , assuming a constant compressiblity factor in the pressure-density relation. The oscillations are due to roughness on the bounding surfaces of the fluid film. As shown by previous studies, homogenization is an effective approach for analyzing the effects of surface roughness in hydrodynamic lubrication. By two-scale convergence theory we obtain the limit problem (homogenized equation) and strong convergence in L2 for the unknown density ρε . By adding a small corrector term we also obtain strong convergence in the Sobolev norm.. 1. Introduction Lubrication problems involve surfaces which are close to each other and in relative motion. Such examples include bearings, hip joints and gearboxes. To reduce friction and wear, a thin film of viscous fluid (lubricant) is injected between the surfaces. In order to understand and optimize the effects of lubrication it is important to describe the flow in the lubricant film. For many applications, Reynolds’ lubrication equation, relating the pressure and density, gives a good approximation of the flow. When either of these are known it is possible to compute other fundamental quantities (velocity field, friction force, load carrying capacity etc.). The friction force gives information about the force needed to keep the surfaces in relative motion. Integrating the pressure over one of the surfaces yields the transversal load carried by the surface (load carrying capacity). In many applications the distance between the surfaces is so small that the surface roughness has to be taken into account. The main focus of this paper is to model and analyze the effects of surface roughness under the assumption that the fluid has constant compressibility (see (2) below). 2000 Mathematics Subject Classification. 35J60, 35J65, 35B27, 35Q35, 76D08. Key words and phrases. homogenization, multiscale convergence, two-scale convergence, hydrodynamic lubrication, Reynolds equation, surface roughness. 1.

(60) 2. A. ALMQVIST, J. FABRICIUS, AND P. WALL. We begin with describing the fluid film separating two rigid surfaces. At time t = 0 we assume that the film is bounded by the surfaces x3 = h+ (x1 , x2 ) (the upper surface contained in the region x3 > 0) and x3 = h− (x1 , x2 ) (the lower surface contained in the region x3 < 0), where h+ and h− are functions defined on R2 . For simplicity we assume that the motions of the surfaces are translational with constant velocities and parallel to the plane x3 = 0. The corresponding velocity vectors are denoted by V + = (v1+ , v2+ ) and V − = (v1− , v2− ). Given a bounded domain Ω in R2 , we assume that the region in space . Ft = (x, x3 ) : x = (x1 , x2 ) ∈ Ω, h− (x − V − t) < x3 < h+ (x − V + t) is filled with fluid for all t ∈ [0, T ]. If the fluid film Ft is thin, the pressure p and density ρ in the fluid film are approximately governed by the Reynolds equation [27]: h3 ρ hρ + ∂ − (hρ) + div − ∇p + (V + V ) = 0 in Ω × (0, T ], (1) ∂t 12µ 2. where µ is the viscosity of the fluid and h is the function defined by h(x, t) = h+ (x − tV + ) − h− (x − tV − ). In this sense h describes the thickness of the film. For a derivation of the Reynolds equation see e.g. Hamrock [19], where it is also shown how the velocity field of the fluid is recovered from p. Clearly equation (1) must be complemented with some relation between the unknowns p and ρ. The simplest case is obtained if one assumes that ρ is constant throughout the film. Then we obtain the incompressible Reynolds equation ∂h h3 h + − + div − ∇p + (V + V ) = 0, ∂t 12µ 2 which has only one unknown p. In this article we analyze a case where ρ is not constant. As a measure of the compressibility of a fluid one can introduce the compressibility factor β, which is defined as the relative change of the density with respect to a pressure change, dρ/dp = βρ. In general β depends on both pressure and temperature, but if we assume β is constant we obtain the density-pressure relation (2). ρ = ρa eβp ,.

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