Two-scale homogenization of a hydrodynamic Elrod-Adams model

Transcription

1 March Two-scale homogenization of a hydrodynamic Elrod-Adams model G. Bayada MAPLY CNRS UMR-5585 / LAMCOS CNRS UMR-5514, INSA Lyon, Villeurbanne cedex, France S. Martin MAPLY CNRS UMR-5585, INSA Lyon, Villeurbanne cedex, France C. Vázquez Dep. Matemáticas, Universidade A Coruña, Campus Elviña, A Coruña, España ABSTRACT The present paper deals with the analysis and homogenization of a lubrication problem, via two-scale convergence. We study in particular the Elrod-Adams problem with highly oscillating roughness effects. 0 Statement of the problem Cylindrical thin film bearings are commonly used for load support of rotating machinery. Fluid film bearings also introduce viscous damping that aids in reducing the amplitude of vibrations in operating machinery. A plain cylindrical journal bearing is made of an inner rotating cylinder and an outer cylinder. The two cylinders are closely spaced and the annular gap between the two cylinders is filled with some lubricant. The radial clearance is very small, typically ÖÖ for oil lubricated bearings. The smallness of this ratio allows for a Cartesian coordinate to be located on the bearing surface. Thus, the Reynolds equation has been used for a long time to describe the behaviour of a viscous flow between two close surfaces in relative motion (see [37, 38] for historical references). The transition of the Stokes equation to the Reynolds equation has been proved by Bayada and Chambat in [11]. In dimensionless coordinates, it can be written as Ö ÖÔ where Ô is the pressure distribution, and the height between the two surfaces. Nevertheless, this modelling does not take into account cavitation phenomena: cavitation is defined as the rupture of the continuous film due to the formation of air bubbles and makes the Reynolds equation no longer valid in the cavitation area. In order to make it possible, various models have been used, the most popular perhaps being variational inequalities which have a strong mathematical basis but lack physical evidence. Thus, we use the Elrod-Adams model, which introduces the hypothesis that the cavitation region is a fluid-air mixture and an additional unknown (the saturation of fluid in the mixture) (see [22, 24, 25, 28]). The model includes a modified Reynolds equation, here referred exact Reynolds equation with cavitation (see problem È µ in the next section). From a mathematical point of view, the problem can be simplified using a penalized Reynolds equation with cavitation (see problem È µ in the next section). Homogenization process for lubrication problems is mainly related to the roughness of the surfaces. Let us mention that the Reynolds equation is still valid as long as, being a small parameter describing the roughness spacing, and being the film thickness order (assumed to be small too) (see [12] for details). The study of surface roughness effects

2 2 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model in lubrication has gained an increasing attention from 1960 since it was thought to be an explanation for the unexpected load support in bearings. Several methods have been used in order to study roughness effects in lubrication, the most popular perhaps being the flow factor method (see [35, 36, 41]), which is based on a formulation that is close to the initial one, only modified by flow factors related to anistropic and microscopic effects. So far this procedure has been used either by considering that no cavitation phenomena occur or using variational inequation models. Let us mention that the homogenization of cavitation models using variational inequalities has been studied in [16]. Recently many papers have discussed cavitation phenomena coupled with roughness effects, in mechanical engineering: A generalized computational formulation, by Shi and Salant [40], has been applied to the rotary lip seal and used to predict the performance characteristics over a range of shaft speeds. Interasperity cavitation has been studied in particular by Harp and Salant in [30] in order to derive a modified Reynolds equation with flow factors describing roughness effects and macroscopic cavitation. Modelling of cavitation has been pointed out in particular by Van Odyck and Venner in [42] in order to discuss the validity of the Elrod-Adams model and the formation of air bubbles leading to cavitation phenomena. The above papers are based on averaging methods taking into account statistic roughness and are mainly heuristic. Our purpose, in the present paper, is to study in a rigorous way the limit of a three dimensional Stokes flow between two close rough surfaces using a double scale asymptotic expansion analysis (see for instance [14]) in the Elrod-Adams model. The paper is organized as follows: Section 1 is devoted to the mathematical formulation of the lubrication problem: we briefly present the exact Elrod-Adams problem along with its penalized version. We also give the existence and uniqueness results corresponding to each problem. For this, we use a well-known penalization method to get the existence result. Uniqueness of the pressure is obtained using the doubling variable method of Kružkov, which has been extended by Carrillo to the dam problem. Section 2 deals with the homogenization process: after some preliminaries on the twoscale technique, we first establish an uncomplete form of the homogenized problem in which an additional term in the direction perpendicular to the flow but also anisotropic phenomena on the saturation appear. In order to complete the homogenized problem, we introduce additional assumptions that lead us to consider particular but realistic cases: considering a separation of the microvariables on the gaps allows us to completely solve the difficulties previously mentioned; then, taking into account oblique roughness, we show that we obtain an intermediary case between the uncomplete problem (general case) and the complete problem (with the separation of the microvariables). Section 3 presents the numerical method and results which illustrate the main theorems established in the previous sections: we study longitudinal and transverse roughness cases.

3 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 3 1 Mathematical formulation 1.1 The lubrication problem The dimensionless domain is denoted and we suppose that the following assumptions are satisfied: Assumption 1.1 µ is periodic and satisfies Ü Üµ Assumption 1.2 Ô is a Lipschitz continuous non-negative function, periodic. Now let us introduce the Elrod-Adams model taking into account cavitation phenomena. Thus we introduce an exact problem and a penalized problem. µ Exact Reynolds problem - The strong formulation of the problem is given by the following set of equations: Ö ÜµÖÔ Üµ ܵ ܵ Ü Ô Üµ Ô Üµ ܵµ ܵ Ü with the following boundary conditions: Ô on and Ô Ô on, Ô and Ô are periodic, (Dirichlet conditions) (periodic conditions) where ܵ is the normalized height of fluid between the two surfaces. The boundaries and are given on FIG.1. These boundary conditions are linked with a specific but wide type of bearings: journal bearings with a pressure imposed on the top and at the bottom. However, other boundary conditions can be considered. The earlier problem can be formulated under a weak form as Ü Figure 1: Normalized lubrication domain (with supply pressure) È µ Find Ô µ Î Ä µ such that: ÖÔÖ Î Ô Ô µ a.e. in,

4 4 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model where the functional spaces are defined as Î Î Ò Ò À µ is periodic À µ is periodic Ô Ó Ó µ Penalized Reynolds problem - In the penalized problem, an approximate relationship between Ô and is used. Defining the function À Þµ if Þ, Þ if Þ, if Þ, the weak formulation of the problem is given by È µ Find Ô Î, such that: ÖÔ Ö Ô a.e. in. À Ô µ Î Hence, À Ô µ plays the role of the saturation function. Let us mention that, by many aspects, the lubrication problem is close to the dam problem. The dam problem has first been stated using variational inequalities (see [7, 8, 9, 17]). But this approach is only possible for dams with vertical walls (typically rectangular dams). The formulation of the dam problem for domains with general shapes has been introduced By Alt, Brézis, Kinderlehrer and Stampacchia [2, 21]. Introducing the permeability of the porous medium, denoted, the formulation is based on Darcy s law ([26] for historical references). The basic problem is to find the pressure Ô and the fluid saturation in the domain. The main differences with the lubrication problem lie in the flow direction ( in the lubrication problem, Ü in the dam problem) and an additive sign condition on the fluid flow in the dam problem, designed to eliminate the non physical solutions and meaning that no water flows into the dam through the boundary in contact with the open air. Homogenization of the dam problem using the -convergence has been partially studied by Rodrigues (see [39] and related references). 1.2 Existence and uniqueness results for È µ Let È Ò µ be the auxiliary problem defined by È Ò µ Find Ô Ò Î such that, Ô Ò Î being given, ÖÔ Ò Ö À Ô Ò µ Î Lemma 1.3 Under Assumptions 1.1 and 1.2, problem È Ò µ admits a unique solution Ô Ò. Moreover, one has the following estimates: where C does not depend on Ò. Ô Ò À µ

5 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 5 Proof. Equivalently, with Õ Ò Ü µ Ô Ò Ü µ Ô Üµ ܵ (with ܵ Ü for example), one has to find Õ Ò Î such that ÖÕ Ò Ö À Ô Ò µ Ö Ô µö Î Existence and uniqueness are consequences of Lax-Milgram s theorem. Estimates are obtained using Õ Ò as a test function and Cauchy-Schwartz inequality, trace theorem and Poincaré- Friedrichs inequality. Theorem 1.4 Under Assumptions 1.1 and 1.2, problem È µ admits a unique solution Ô. Proof. Existence of a solution is obtained by studying the behaviour of Ô Ò when Ò goes to. By estimates of Lemma 1.3, there exists Ô À µ such that, up to a subsequence, Consequently, for every Î. Ô Ò Ô in À µ. ÖÔ Ò Ö ÖÔ Ö As À µ Ä µ with compact injection and À is Lipschitz continuous, one has for every Î. Then one has: À Ô Ò µ ÖÔ Ö Moreover, by Theorem III.9 of [20], À Ô µ À Ô µ Î (1) Ô Î (2) From Equations (1) and (2), we deduce that Ô is a solution of È µ. Positivity of solutions is obtained by rewriting Ô as Ô Ô Ô Ô Ô ÑÜ Ô µ ÑÒ Ô µ with It can be proved that Ô Î. Using Ô as a test-function in the variational formulation (1), one has Then Ô a.e. and Ô a.e. in. ÖÔ Uniqueness of the solution is obtained using a particular test function (following an idea developped in [10]). Let Ô and Ô be two solutions of È µ. Then Õ Ô Ô satisfies: ÖÕÖ À Ô µ À Ô µ Î (3)

6 6 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model We consider the test function Æ Õµ, where Æ is defined with the usual notation for the positive part of a function by Æ Üµ Æ Ü if Ü, if Ü. Since Æ is Lipschitz continuous, Æ Õµ Î (see [29]). Moreover, one has Ö Æ Õ ÕÆ ÖÕ where is the characteristic function, defined to be identically one on and zero elsewhere. From Equation (3) and Assumption 1.1, we deduce: Then it follows: Ü Õ ÜµÆ ÖÕ Ö ÐÒ Æ Õ Ü Õ ÜµÆ Õ Æ Õ Õ Æµ Æ Applying Poincaré s inequality we obtain: ÐÒ Ü Õ ÜµÆ Õ Æµ Æ À Ô µ À Ô µ ÕÜ Õ ÐÒ Ö ÐÒ Õ Æµ Æ Õ Æµ where depends on,, and but does not depend on Æ. Then letting Æ, Õ Üµ a.e. in. Exchanging the roles of Ô and Ô gives Õ Üµ a.e. in so that, finally, Õ Ô Ô a.e. in. Æ Æ 1.3 Existence and uniqueness results for È µ Theorem 1.5 Under Assumptions 1.1 and 1.2, problem È µ admits at least one solution. Proof. Existence of a solution is obtained by studying the behaviour of Ô when goes to 0. First, let us notice that the following estimates hold: Ô À µ À Ô µ Ä µ where and do not depend on. Indeed, they are easily obtained by considering the properties of À and using Ô Ô as a test function. From the earlier estimates, one has:

7 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 7 (i) Ä µ, À Ô µ, in Ä µ weak-. In particular, À Ô µ Î (ii) Ô À µ, Ô Ô, in À µ and Ô Ô, in Ä µ. In particular, From µ and µ, we deduce ÖÔ Ö ÖÔ Ö Ü ÖÔÖ À Ô µ Î Î Moreover, considering Theorem III.9 of [20], Ô Î. It remains to prove the following properties to complete the proof of existence of a solution for the initial problem È µ: µ Ô a.e. in, µ a.e. in, µ Ô µ a.e. in. Proof of µ is deduced from positivity of Ô (see Lemma 1.4) and strong convergence of Ô to Ô in Ä µ. Proof of µ is obtained considering the properties of the weak- convergence (see Proposition III.12. in [20]). Since we have then, ÐÑ Ò À Ô µ Ä µ À Ô µ in Ä µ weak-, Ä µ, and finally, a.e. in. Let us prove that a.e. We settle À Ô µ. We have Ä µ and Ä µ in Ä µ weak-. The weak- topology is separated. Then and we have the following property: which can be rewritten as ÐÑ Ò Ä µ Ä µ Ä µ i.e. a.e. in. Proof of µ is obtained with the following method: let À denote the Heaviside graph. Since Ô (see Lemma 1.4), the following property holds: À Ô µ Ô From this, we have Ô À Ô µµ Ô À Ô µ À Ô µµ. This term is analyzed in two steps:

8 8 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 1st step - Let be a function in Ä µ. Then, Ô À Ô µµ Ô µ Using Cauchy-Schwarz inequality, Ô À Ô µµ Ô µ Ôµ À Ô µµ Ô Ô Ô Ä µ Ä µ Ô À Ô µµ Ô À Ô µµ With the Ä strong convergence of Ô to Ô and the weak- convergence of À Ô µ to, since Ô Ä µ, we get We have proved that Ô À Ô µµ Ô µ Ô À Ô µµ Ô µ in Ä µ. 2nd step - Let be a function in Ä µ. Then, by construction of À, Ô À Ô µ À Ô µ Ô Ô with Ü Ô Üµ. We have proved that À Ô µ À Ô µ Ô in Ä µ. From uniqueness of the weak limit in Ä µ and the results stated in the two previous steps, we deduce: Ô µ in Ä µ. We state a uniqueness result following an idea widely developped by Alvarez and Oujja in [5] for the unstationary case. The uniqueness result is based on a monotonicity result when comparing the value of two solutions on the upper boundary. Thus we first establish the following lemma: Lemma 1.6 Let Ô µ and Ô µ two solutions of (È ) with respective pressure boundary values Ô and Ô on. Then, Proof. 1st step: Test functions Let Ü µ and Ü Ü function: Ü Ô Ô µ µ Ü µ Ü Ý µ Ü Ü Ü Ü µ µ be two pairs of variables and let us define the following Ü Ü

9 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 9 where µ, Öµ Ö supports in µ., Öµ Ö. and are functions with If dist Supp µ, then the functions µ and µ vanish on the boundary (see [4] for the details and [5]). Moreover, in order to get a periodic function, we choose an even function and redefine it when Ü µ belongs to the subset by setting Ì Ë Ò Ü µ Ü Ü Ó Ü Ü Ü Ü Then we define the following function: Ô µ Ô µµ µ ÑÒ µ Thus, for fixed (resp. ), µ (resp. µ belongs to Î. 2nd step: Integral equality Let us denote and Ö (resp. and Ö ) the domain and the gradient vector for the variable (resp. ). For fixed, let us use µ as a test function in the variational formulation of È µ with the variable : µ Ö Ô µ Ö µ µ µ µ Integrating the previous equation on gives us a first integral equality on É. Applying the same method to the variable (and exchanging the roles of and ), we get a second integral equality. Then from periodicity and boundary conditions, it is possible to establish: É µ Ö Ö µ Ô 3rd step: Change of variables We make the following change of variables: The integral equality becomes: ÉÞ Þ µö Þ É Ô Þ µ µ Ö Ö µ Ô µ µ µ Þ Ö Ö µ Ü µ Þ µö Þ Ô Þ µ ÖÞ Þ Þ µ Þ ÉÞ Þ µ Þ µ Þ µ Þ Þ µ Þ where É Þ is the image of the domain É through the change of variables. Þ

10 10 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Let us consider the sets: Ò Þ µ É Þ Ô Þ µ Ô Þ Ó µµ Þ Þ µ Ò Þ µ É Þ Ô Þ µ Ô Þ Ó µµ Þ Þ µ Let us denote Á (resp. Á ) the contribution of (resp. ) in the first integral and let us denote  (resp.  ) the contribution of (resp. ) in the second integral. Then we have: Á Á  Â. 4th step: Study of the integrals Let us study  : since does not depend on Þ, one gets: Â. Let us study  : Â Þ µ Þ µ Þ µ Þ µ Ô Ô µ Þ Ô Ô µ Þ Þ Þ Ô Ô µ Þ µ Þ Þ The first integral can be rewritten as Ô Ô Â µ Þ µ Þ µ ÑÒ Þ Þ É Þ Integrating by parts, letting, and using Lebesgue theorem, we get: ÐÑ Â Þ µ Þ µ É Þ Þ Þ Since Supp µ, Supp µ and we get: ÐÑ Â µ É Þ Þ µ µ µ, and finally ÐÑ ÐÑ Â The second integral can be rewritten in the old variables as Â Ô Ô µ É Ü É µ Ô Þ since when Ô. Rewriting the integral, one gets: Â Ô µ ÑÒ É µ µ Ô Ô Þ µ µ µ Þ is a Lipschitz continuous function, Ô Ô Þ

11 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 11 using the Green formula with periodicity and boundary conditions. Since the function µ µ µ is bounded for each,, we conclude and finally, Let us study Á : Á By Lebesgue theorem, ÐÑ Á ÐÑ Â ÐÑ ÐÑ ÐÑ Â Þ µ Ö Þ Ô Þ µ Ö Þ Ô ÖÞ Þ µ µ µ ÉÞØ Þ Ô µ Þ ÉÞØ Þ µ Ô Þ Ô Ô Þ µ µ µ Þ µ µ Þ Ô Ô Ô Þ Þ µ µ µ É ÞØ Þ µ Ô Ô Þ Ô Ô Þ µ µ µ Using the properties of, and since is a Lipschitz continuous function, it is easy to conclude that the first integral goes to 0 when. Then we obtain, studying the behaviour of the second integral (see [4] for the details): ÐÑ Á Let us study Á : Rewriting Á in the old variables gives: Á µ Ö Ô Üµ Ô Ô Ü µ Ü Ü µ µ Ô Ö Ô Ö Ö Ô The first integral is positive. The second integral satisfies: µ Ô Ö Ô Ö µ Ô Ô Ö Ô Ö µ Ö Ô Ö É µ Ö Ô Ö µ Ö Ô Ö µ Ö Ô Ö É É µ Ö Ô Ö Ô µ Ö Ô Ö µ Ö Ô Ö

12 12 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model By Hölder inequality and since ÐÑ, one gets: ÐÑ In a similar way, ÐÑ and we deduce µ Ô Ö Ô Ö Ô µ Ö Ô Ö ÐÑ µ É ÐÑ µ É Ö Ô Ö Ö Ô Ö ÐÑ Á Now passing to the limit ( ) in the integral equality concludes the proof. Theorem 1.7 Let Ô µ and Ô µ two solutions of (È ) with respective pressure boundary values Ô and Ô on. Let us suppose that Ô Ô. Then Ô Ô a.e. in. Proof. From Lemma 1.6, denoting Ô Ô µ, we have, for every µ, ܵ Ü µ Ü Ü Then one gets: Ü µ Ü Using the following notations: Ü Ü µ Ü µ we get: Ü µ Ü µ Ü µ Ü µ Ü µ Ü Ü µ Ü µ Ý Ü µ Ü µ Ü µ Ü µ (4) Now let us suppose that Ü µ, Ü Ý Ý µ µ and let be a solution of the two points boundary problem: Ü µ Ü µ Ü µ Ü µ Ü µ Ý µ Ý µ (5) where Ý Ý satisfying Ü µ Ü Ý Ý. From the minimum principle, Ü µ Ü Ý Ý. Then we define a regularizing function on Ý Ý such that is a test function for Equation (4) and Ü µ on Ý Æ Ý Æ. More precisely, let Æ be a positive parameter and the function defined on Ý Ý by Ü µ Ü Ý Æ Ü Ý Ý Æµ Ü Ý Æ Ý Æµ Ü Ý Æ Ü Ý Æ Ý Æµ Ý Ü Ü Ý Æ Ý Æµ Æ Ý Ü Æ Ü Ý Æ Ý µ

13 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 13 This function satisfies Ý µ Ý µ and Ý µ Ý µ. Ü µ Ü µ, Ü Ý µ. Therefore, we can take in Equation (4) and get Let be Ü µ Ý Ý. We have Ý Ý µ, Ý µ Ý µ and Ý µ Ý Ý Ü µ Ü µ Ü µ Ü µ Ü By separating the integration intervals, we decompose this integral in the form Ý Æ Ý Ü µ µ Ü µ µ Ü Ý Æ Ý Ý Æ Ý Æ Ü µ Ü µ Ü Ü µ µ Ü µ µ Ü (6) From (5), the second integral is strictly negative, and for the two other integrals, we have Ý Æ Ü µ µ Ü µ µ Ü Ý Ý Æ Ý Ý Æ Ý Ü µ µ Ü µ µü Ü µ Ü µ Ü µ Ü µ Ü µ µ Ü (7) Since Ü µ Æ, Ü µ Æ and being the functions and continuous in the interval Ý Ý Æµ, the terms under the last integral in (7) are bounded and we obtain Ý Æ Ý In the same way, we have Ý Ü µ µ Ü µ µ Ü Æ Ý Æ Ü µ µ Ü µ µ Ü Æ Passing to the limit (Æ ) in inequality (6), one gets: Ý Ý Ü But we have also: Then we have: Ü µ Ý Ý Ý Ü Ý on µ, that is we conclude Ô Ô a.e. in. Ô Ô µ, and Theorem 1.8 Under Assumptions 1.1 and 1.2, problem È µ admits at least one solution Ô µ whose pressure Ô is unique. Moreover, if there exists a set of positive measure where Ô Ü µ, for any Ü, then the saturation is unique.

14 14 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Proof. Uniqueness of the pressure is obtained from Theorem 1.7. Let us consider Ô µ and Ô µ two solutions. Then we get, by means of substraction: µ Î and µ in µ, so that µ is a function only depending on the Ü variable, almost everywhere in. In particular, if there exists a set of positive measure where ܵ ܵ, for every Ü, then a.e. in. We give a supplementary result : Corollary 1.9 Under Assumptions 1.1 and 1.2 and if can be written under the form Ü µ µ Ü µ (with Ü µ ), then problem È µ admits a unique solution. Proof. By Theorem 1.8, it is sufficient to prove that, for any Ü, there exists a set of positive measure, where Ô Ü µ. Let be a test function only depending on Ü. Then we have Thus, we get Ô Ü i.e. Ü µ Ô Ü Ü µ Ü µ Ü Ô Üµ ܵ Ü where is a real constant. Since can be written under the form Ü µ µ Ü µ, dividing the previous equality by Ü µ gives µ Ô Ü µ Ü Ü µ Integrating the previous equality and taking into account the boundary conditions on the pressure, µ Ô Ü µ Ô Ü Øµ Ø Ü (8) We deduce from Equation (8) that, for any Ü, there exists a set of positive measure, where Ô Ü µ. The next sections deal with homogenization of the lubrication problem, using two-scale convergence techniques which have been introduced by Nguetseng in [34], and further developped by Allaire [1], Cioranescu, Damlamian and Griso [23] and Lukkassen, Nguetseng and Wall [31].

15 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 15 2 Homogenization of the lubrication problem In the whole section, and. Now we introduce the roughness of the upper surface; the roughness is supposed to be periodic, characterized by a small parameter denoting the roughness spacing. Due to the shape of the Reynolds equation, oscillating data appear in both sides of the equation. So we are led to consider the following problem È µ and assumptions: Assumption 2.1 Let and be functions such that: (i) Ä µµ or Ä µµ, (ii) Ä µµ or Ä µµ, (iii) Ñ Ü Ýµ Ñ Ü Ýµ, (iv) Ñ Ü Ýµ Ñ Ü Ýµ. We introduce the following functions defined on : ܵ Ü Ü Then we introduce the following problem: È µ ܵ Find Ô µ Î Ä µ such that: ÖÔ Ö Ü Ü Î Ô Ô µ a.e. in. Existence and uniqueness results have been discussed in Section 1. Our purpose is to discuss the behaviour of problem È µ when goes to 0, using two-scale convergence techniques. 2.1 Preliminaries to the two-scale convergence technique First we recall some useful definitions and results for the two-scale convergence (see [1, 23, 31]). Lemma 2.2 The separable Banach space Ä µµ is dense in Ä µ. Moreover, if Ä µµ, then Ü µ ܵ Ü Üµ is a measurable function such that µ Ä µ Ä µµ Definition 1 The sequence Ù Ä µ two-scale converges to a limit Ù Ä µ if, for any Ä µµ, one has ÐÑ Ù Üµ Ü Ü Ü Ù Ü Ýµ Ü Ýµ Ý Ü Lemma 2.3 Let Ù be a bounded sequence in Ä µ. Then there exists Ù Ä µ such that, up to a subsequence, Ù two-scale converges to Ù. Lemma 2.4 Let Ù be a bounded sequence in À µ, which weakly converges to a limit Ù À µ. Then Ù two-scale converges to Ù and there exists a function Ù Ä À µêµ such that, up to a subsequence, ÖÙ two-scale converges to ÖÙ Ö Ý Ù.

16 16 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 2.2 Two-scale convergence results In this subsection, Ô µ denotes a solution of problem È µ. Lemma 2.5 There exists Ô Î such that, up to a subsequence: Ô Ô in À µ and Ô Ô in Ä µ. We have also the following two-scale convergences: (i) Ô two-scale converges to Ô. Moreover, there exists Ô Ä À µêµ and a subsequence, still denoted, such that ÖÔ two-scale converges to ÖÔ Ö Ý Ô. (ii) There exists Ä µ and a subsequence, still denoted, such that twoscale converges to. Moreover, Ô a.e. in. Proof. Since, is bounded in Ä µ and in Ä µ, so that Ä µ, where only depends on. Moreover, from Assumptions 2.1 µ Úµ, properties of and the Cauchy-Schwarz inequality, we get the estimates on Ô by using Ô Ô (with Ô a regular function such that Ô Ô Î ) as a test function and Poincaré-Friedrichs inequality so that Ô À µ where only depends on. The convergence results are the consequence of the previous estimates (see Lemmas 2.3 and 2.4, or Proposition 1.14 in [1], Theorem 13 in [31]). Finally Ô a.e. in due to the properties of Ô. Now, we give the properties of the two-scale limits Ô and, which are quite similar to the ones of the initial functions Ô and. These properties are obtained by means of two-scale convergence techniques. Proposition 2.6 a.e. in. Proof. Let us introduce the classical notation Û ÑÜ Û µ and Û ÑÒ Û µ, for any Û Ä µ. Since Ä µµ is dense in Ä µ (see Theorem 3 in [31]), let us consider a sequence Ò Ä µµ, Ò, which strongly converges to in

17 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 17 Ä µ (note that such a sequence exists 1 ). Thus, defining the following sequences Ò Üµ Ò Ü Ü Ü Ò Ü Ýµ Ò Ü Ýµ Ý Ü we have, using the two-scale convergence of, ÐÑ Ò Ò Obviously, Ò is a sequence of positive numbers so that we have also: Ò. Now letting Ò, we have: ÐÑ Ò Ò µ µ Thus, Ò being a sequence of positive numbers, so that, finally,. Thus, we have proved that a.e. Similarly, it can be proved that µ a.e. Proposition 2.7 Ô µ a.e. in. Proof. By uniqueness of the two-scale limit (see [1, 31]), it is sufficient to prove that Ô µ two-scale converges to Ô µ. As Ô two-scale converges to Ô, let us prove that Ô two-scale converges to Ô. The sequence Ô is bounded in Ä µ. Consequently, it remains to prove (see Proposition 1 in [31]): Ô Üµ ܵ Ü Ü Ü Ô Üµ Ü Ýµ Ü Ýµ Ý Ü for all µµ. Let be a function in µµ and let «be defined by: «Ô ܵ ܵ Ü Ü Ü Ô Üµ Ü Ýµ Ü Ýµ Ý Ü Our purpose is to prove that «tends to 0. Then we have: 1 Let Ä µ,. By Theorem 3 in [31], there exists a sequence Ò Ä µµ such that Ò strongly converges to in Ä µ. Now it is sufficient to prove that (i) Ò Ä µµ, (ii) Ò strongly converges to in Ä µ up to a subsequence. We have the following characterization of Ä µµ (see Theorem 1 of [31]): a function belongs to Ä µ if and only if there exists a subset of measure zero in such that: (a) for any Ü Ò, the function Ý Ü Ýµ is continuous and periodic, (b) for any Ý, the function Ü Ü Ýµ is measurable, (c) the function Ü ÙÔ Ý Ü Ýµ has finite Ä µ norm. Thus, it is obvious that if Ò Ä µµ, then Ò Ä µµ. It remains to prove that, up to a subsequence, Ò strongly converges to in Ä µ. Thus, by Theorem IV.9 in [20], as Ò Ä µ with Ò Ä µ, there exists a subsequence Ò such that (a) Ò a.e. in, (b) Ò Ü Ýµ Ü Ýµ, for all Ò, a.e. in, with Ä µ. Now, since Ò a.e. on and Ò Ü Ýµ Ò Ü Ýµ Ü Ýµ we state from the Lebesgue theorem that Ò Ä µ, and the proof is concluded.

18 18 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model «Ô ܵ Ô Üµ ܵ Ü Ü ßÞ Ð Ô Üµ ܵ Ü Ü µ Ü ßÞ Ð Ü Ô Üµ Ü Ýµ Ü Ýµ Ý Ü Using the Cauchy-Schwarz inequality and Lemma 2.2 (see also Lemma 1.3 in [1] or Theorem 3 in [31]), we have: As Ô Ô in Ä µ, we have: Ô Ô Ä µ µ Ô Ô Ä µ Ä µ. Ä µµ In order to prove that, since two-scale converges to, it is sufficient to prove that Ü Ýµ Ü Ýµ Ô Üµ Ü Ýµ is an admissible test function for the two-scale convergence (i.e. Ä µµ). Let us prove that Ü Ýµ Ô Üµ Ü Ýµ Ä µµ for every µµ. With µµ and Ô À µ, we have for a.e. Ü in : Ô Üµ Ü µ µ µ Let us denote Ü Ýµ Ô Üµ Ü Ýµ. As Ô À µ Ä µ, Ä µµ, by the Cauchy-Schwarz inequality, Ä µµ Ô Üµ ÙÔ Ü Ýµ Ü Ý Ô Üµ Ü ÙÔ Ý Ü Ýµ Ü µµ We have proved that Ü Ýµ Ô Üµ Ü Ýµ Ä µµ for any function µµ. Then,. 2.3 Homogenization of the lubrication problem (general case) Using an idea developped in [1], one has the following macro-microscopic decomposition: Theorem 2.8 From the initial formulation, Macroscopic equation: for every in Î. ÖÔ Ö Ý Ô Ý Ö Ü Ý Ü (9)

19 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 19 Microscopic equation: For a.e. Ü, for every À µ. ÖÔ Ö Ý Ô ÖÝ Ý Ý Ý (10) Proof. Using the test function ܵ ܵ Ü with Î, µ and À µ in problem È µ, one has: Ü Ü ÖÔ Üµ Ü Ö Üµ ÜµÖ Ý Üµ Ü Ü Ü Üµ ܵ Ý Ü ÖÜ Üµ Ü Ü Üµ Passing to the limit ( ) gives us the macroscopic equation (with Ü ) and the microscopic equation (with ), using density results. Let us define the local problems, respectively denoted µ, Æ µ and Æ µ: Find Ï,, ( ) in Ä À µêµ, such that, for almost every Ü : We immediatly have: Ö Ý Ï Ö Ý Ö Ý Ö Ý Ö Ý Ö Ý Lemma 2.9 Problem µ (resp. Æ µ, Æ in Ä À µêµ. Theorem 2.10 The homogenized problem can be written as with È µ coefficients defined as À µ µ (11) Ý À µ µ (12) Ý À µ µ (13) Ý µ) admits a unique solution Ï (resp., ) Find Ô µ Î Ä µ Ä µ such that ÖÔ Ö Ö Î Ô and Ô µ µ a.e. in,, Æ Ï Ý and ܵ Ü Ýµ Ý, being the homogenized Moreover, the homogenized problem admits at least a solution. Ý

20 20 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Proof. From Lemma 2.9, one has: with Ï Ô Ü Ýµ Ï Ü Ýµ ÖÔ Üµ Ü Ýµ in Ä À µêµ (14) Ï. Let us notice that Ü Ýµ depends on Ü Ýµ which is unknown. Ï Using Equation (14) in the macroscopic equation gives: Á ÖÏ ÖÔ Ö for every Î. Introducing the notations µ with ÖÔ Ö Á ÖÏ µ Ý Ý Ü Ö ( ), one gets Ý Î Introducing the ratios in the vector concludes the proof. Remark 2.11 The homogenized lubrication problem can be considered as a generalized Reynolds-type problem with two saturation functions ( ). Let us notice that if there is no cavitation phenomena (i.e. Ô ) then : thus, we get the classical homogenized Reynolds equation (without cavitation) (see [13]). But several aspects remain hard to describe: (a) The homogenized problem leads us to consider two different saturation functions, since an extra term has to be added (in the Ü direction of the flow) when comparing the homogenized problem to the initial problem. (b) Another point is to consider the fact that the property is missing, i.e. we cannot guarantee that homogenized cavitation parameters are smaller than 1 in cavitation areas! (c) We are not able prove any uniqueness result, for the homogenized problem, using the methods described in Section 1. (d) Algorithms are known to solve the roughless problem (see for instance the papers by Alt [3], Bayada, Chambat and Vazquez [15], Marini and Pietra [32]). But how to solve the homogenized problem numerically? How to treat the two different saturation functions? Thus these four difficulties have to be underlined in the most general case and, in the following subsections, we show how it is possible to solve them, fully or at least partially. Additional assumptions have to be made in order to get an homogenized problem with a structure which is similar to the initial one. This will be the subject of the following subsection. Before starting this study, let us conclude this subsection with the following theorem: (15)

21 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 21 Theorem 2.12 The homogenized problem È µ admits a solution Ô µ with a.e. in. Proof. The result is obtained in three steps: first, we consider the penalized rough problem µ; then, we apply the homogenization process to the penalized problem (i.e. ); È finally, we pass to the limit on the penalization parameter (i.e. ). 1st step - Let us consider the rough penalized problem: È µ Find Ô Î such that: ÖÔ Ö À Ô µ Ô a.e. in Î 2nd step - Similarly to the exact rough problem, we get a priori estimates on the pressure, i.e. Ô À µ where only depends on. From the previous estimate, we deduce that there exists Ô Î (Ô a.e. in ) such that, up to a subsequence, Ô weakly converges to Ô in À µ. Moreover, Ô two-scale converges to Ô and there exists Ô Ä À µêµ and a subsequence still denoted such that ÖÔ two-scale converges to ÖÔ Ö ÝÔ. Then, with the two-scale homogenization technique, we get the following macro/microscopic decomposition: Macroscopic equation: ÖÔ Ö ÝÔ for every in Î. Microscopic equation: Ý Ö Ü À Ô µ Ý Ü (16) For a.e. Ü, ÖÔ Ö ÝÔ ÖÝ Ý À Ô µ Ý (17) for every À µ. Then introducing the local problems defined in Equations (11) and (12), we get: Ô Ü Ýµ Ï Ü Ýµ ÖÔ Üµ À Ô Üµµ Ü Ýµ in Ä À µêµ. (18) Using Equation (18) in the macroscopic equation gives: Á ÖÏ ÖÔ Ö À Ô µ Ý À Ô µ Ý Ü (19) for every Î. Then, using the definitions of ( ) (see Theorem 2.10) and introducing vector whose th component is, the homogenized penalized problem can be written as Find Ô Î such that È µ ÖÔ Ö Ô a.e. in. À Ô µ Ö Î

22 22 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 3rd step - As is a coercice matrix (see [18]), we establish a priori estimates on Ô, in the À µ norm, which do not depend on, so that there exists Ô Î, (Ô a.e. in ) and Ä µ such that Ô Ô in À µ, À Ô µ in Ä µ weak-. Passing to the limit ( ) in problem È µ concludes the proof, since the properties and Ô µ a.e. in are classically obtained as in Section 1. Remark 2.13 Let us recall that we are not able to prove a uniqueness result on the general problem. But we can wonder if it is possible to obtain a uniqueness result among the class of solutions Ô µ satisfying with (and, of course, Ô, Ô µ ). In fact, it is not possible to get such a result using the method described in Section 1, because it is not well-suited to a flow whose component in the Ü direction is different from 0. Remark 2.14 Theorem 2.12 guarantees that we are able to build an homogenized problem with isotropic saturation from the penalized problem, although it is not the case when directly studying the homogenization of the exact problem (in the most general case): the penalized problem allows us to build a solution in pressure/saturation Ô µ where the saturation satisfies (and, also, Ô and Ô µ ); by contrast, the exact problem with the homogenization process builds a solution in pressure / double-saturation Ô µ for which we are not able to conclude that (although the following properties hold: Ô and Ô µ, µ). At that point, it is important to know whether Ü Ýµ depends on Ý or not: that does not depend on the Ý variable would mean that the homogenized exact problem and the homogenized penalized problem (after passing to the limit on ) are identical, i.e. saturation phenomena would be isotropic. More precisely, in the exact homogenized problem, such an assumption leads us to (see Equations (13) and (19)), (see Propositions 2.6 and 2.7). But, in fact, numerical tests evidence that such an assumption is not valid in general, as it will be pointed out in the next section. Remark 2.15 It is now possible to find, numerically, a solution of problem È µ, by focusing on solutions Ô µ satisfying (with Ô and Ô µ ), and using algorithms that have been previously mentioned. In that prospect, it allows us to eliminate another difficulty that has been underlined in Remark But, since we do not have any uniqueness result, we cannot guarantee that each solution Ô µ satisfies and we are not able to build numerically solutions with two different saturation functions. We can neither illustrate numerically anisotropic effects on the saturation, nor prove that all the solutions have the form Ô µ.

23 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Some particular cases Longitudinal and transverse roughness Our interest in studying the behaviour of the solution when considering transverse or longitudinal roughness is highly motivated by the mechanical applications. From a mathematical point of view, we may even consider a product of transverse and longitudinal roughness i.e. we should consider, in this subsection, the following assumption: Assumption 2.16 (i) Ü Ýµ Ü Ý µ Ü Ý µ, (ii) Ñ Ñ, ( ), (iii) Ü Ýµ Ü Ý µ Ü Ý µ, (iv) Ñ Ñ, ( ). It is clear that the earlier assumption is just a separation of the microscale variables, which allows us to take into account either transverse or longitudinal roughness effects, but also particular full two dimensional roughness effects. For a dimensionless journal bearing, we may consider gaps with roughness patterns described on FIG.2 5, corresponding to a roughless gap Ó µ, Ü. Lemma 2.17 Under Assumption 2.16, it follows that: Proof. Diagonal terms of the matrix. For this, let us recall the variational formulation (see Equation (11)) of problem µ ( ): Ö Ý Ï Ö Ý À µ Ý Let, with. Denoting the averaging process of a function on the Ý variable and using a test function only depending on Ý, one has: Ï Ý Then, one has, for a.e. Ü, that: Ý Ï Ý À Ý µ Using Assumption 2.16 and dividing Equation (20) by, we have: Ï Ý Üµ (20)

24 24 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model MAX x 2 x 1 MIN Figure 2: Normalized gap (no roughness patterns) MAX x 2 x 1 MIN Figure 3: Normalized gap with transverse roughness patterns MAX x 2 x 1 MIN Figure 4: Normalized gap with longitudinal roughness patterns MAX x 2 x 1 MIN Figure 5: Normalized gap with two dimensional roughness patterns

25 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 25 Now, averaging on the Ý variable and using the periodicity of Ï give us so that ܵ. Moreover, using the definition of (see Theorem 2.10) and Equation (20), one has ܵ ܵ, so that ܵ ܵ Non-diagonal terms of the matrix. For this, let,. Recalling the variational formulation of problem µ ( ) and using a test function only depending on Ý, one has: Ï À Ý Ý µ Then, for a.e. Ü, we have: Ï Üµ (21) Ý Using Assumption 2.16, dividing by, averaging on the Ý variable and since Ï periodic, we get that ܵ (for ). Moreover, using the definition of (see Theorem 2.10) and Equation (21), one has ܵ ܵ so that ܵ µ. Lemma 2.18 Under Assumption 2.16, we deduce that: where the following relationships hold: and Ô µ a.e. in. Moreover, the homogenized coefficient satisfies: ܵ is (22) ܵ (23) Proof. The first part of the proof lies in the determination of vector. In the second part, we calculate the homogenized coefficient. 1st part - Computation of the components of vector : Á Let us study the first term of vector. Thus, denoting Û and combining problems (Æ ) and (Æ ), one gets, for a.e. Ü, that: Ö Ý Û Ö Ý µ À µ Ý Now, using a test function only depending on Ý, one has:

26 26 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Û Ý µ Ý Ý À µ Then, for a.e. Ü, we get: Û Ý µ ܵ (24) where ܵ is an additive constant only depending on Ü. Next, using Assumption 2.16, dividing by, averaging on the Ý variable and using the periodicity of Û, we deduce the following equality µ ܵ Now, from Proposition 2.6 and Assumption 2.16, it is easy to get ܵ. Then, averaging Equation (24) on the Ý variable, we obtain that µ µ i.e. Ý Ý Next, applying the earlier method to the variational formulation of problem (Æ ), it is easy to conclude ( ). Á Let us now study the second term of vector. Applying the same mehod (as earlier) to the variational formulation of problem (Æ ), one has: Ý Ý À µ Then, one gets: Ý in À µê. From the previous equality, one obtains: Ý Üµ (25) for a.e. Ü, where ܵ is an additive constant only depending on Ü. Next, using Assumption 2.16, dividing by, averaging on the Ý variable and using the periodicity of, we get that ܵ. So, from Equation (25), we deduce: i.e. Ý With the earlier method applied to the variational formulation of problem (Æ ), it is easy to conclude that.

27 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 27 Now, since we have proved that and, using the definitions of ( ), it is easy to conclude that Equation (22) and property a.e. in hold. Moreover, property Ô µ a.e. in is obtained from Proposition 2.7 and the definition of. Thus, it remains to calculate the homogenized coefficient. 2nd part - Computation of : First, considering problem (Æ ), one gets: Ö Ý Ö Ý À µ Ý for a.e. Ü. Next, using a test function only depending on Ý, one has: À Ý Ý Ý µ Then, Ý Üµ (26) for a.e. Ü, where ܵ is an additive constant only depending on Ü. Using Assumption 2.16, dividing by, averaging on the Ý variable and using the periodicity of, leads to the following equality: ܵ (27) Next, from the definition of (see Theorem 2.10) and Equation (26), we deduce that ܵ ܵ so that, from Equation (27), we conclude the proof. Lemma 2.19 Under Assumption 2.16, it follows that ܵ ܵ (28) Proof. Notice that can be calculated by using the same method which allowed us to obtain in the proof of Lemma 2.18, just replacing problem (Æ ), by problem (Æ ). Then, we have ܵ ܵ (29) The definition of (see Theorem 2.10), Equations (23) and (29) conclude the proof. To summarize the earlier results, we establish the following homogenized problem: Theorem 2.20 Under Assumption 2.16, the homogenized problem is: È µ Find Ô µ Î Ä µ such that: ÖÔ Ö with the following homogenized coefficients: Î Ô Ô µ a.e. in

28 28 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Moreover È µ admits at least Ô µ as a solution, where ܵ ܵ ܵ ܵ (30) and Ô µ is the two-scale limit of Ô µ (solution of problem È µ). Remark 2.21 In the lubrication problem, Assumption 2.16 implies that the gap between the two sufaces is described by the function: Ü Ü Ü Ü Ü In this case, the homogenized coefficients are the following ones: ܵ ܵ and we get the precise link between the microscopic cavitation and the macroscopic cavitation, i.e. ܵ ܵ (31) Theorem 2.22 µ Under Assumption 2.16, problem È µ admits at least a solution Ô µ. Moreover, the pressure Ô is unique, and if there exists a set of positive measure where Ô Ü µ, for any Ü, then the saturation is unique. µ If can be written under the form Ü µ µ Ü µ, problem È µ admits a unique solution. Proof. For µ, existence of a solution is stated in Theorem 2.20, by means of construction via the two-scale convergence techniques. Uniqueness of the pressure and, under the additional assumption, of the saturation is obtained as in Theorem 1.8. For µ, the result is obtained as in Corollary 1.9. Remark 2.23 A primal naive attempt leading to the homogenized problem would be to determine an equation satisfied by the weak limits of Ô µ, namely Ô µ. Interestingly, the weak limit of the pressure does appear in the homogenized problem, but the macroscopic homogenized saturation is a modified average of, weighted by the roughness effects through the influence of functions.

29 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 29 It is interesting to notice that Assumption 2.16 allows us to solve the four difficulties that we could not overcome in the most general case (see Remark 2.11). In particular, there is one single saturation function; the homogenized problem can be numerically solved using algorithms applied to the roughless problem; and it is easy, under additional realistic assumptions, to obtain a uniqueness result on both pressure and saturation. Moreover, Assumption 2.16 includes some important particular cases in terms of mechanical applications: transverse and longitudinal roughness. The results are easily deduced from Theorem 2.22 and given, in the next results, for a strong formulation. Corollary 2.24 If does not depend on Ý (transverse roughness), then the homogenized problem can be written as: Ô Ô Ü Ü Ü Ô Üµ Ô Üµ ܵµ ܵ Ü with the following boundary conditions: Ô on and Ô Ô on Ô and Ô are periodic (Dirichlet conditions) (periodic conditions) Corollary 2.25 If does not depend on Ý (longitudinal roughness), then the homogenized problem can be written as: Ô Ô Ü Ü Ü Ô Üµ Ô Üµ ܵµ ܵ Ü with the following boundary conditions: Ô on and Ô Ô on Ô and Ô are periodic (Dirichlet conditions) (periodic conditions) Under Assumption 2.16, the homogenized problem is similar to the dependent one, since there is one single saturation function. This assumption, imposing a particular form of the roughness, seems to be strong but it allows us to take into account some two dimensional roughness effects. Moreover, it is somewhat surprising to see that passing from the classical homogenized equation (without cavitation) (see [13]) to the one obtained in our paper (including cavitation) only needs to introduce a saturation in the right hand side; in other terms, comparing the homogenized Reynolds equations - with or without cavitation -, the homogenized coefficients are not modified, although the Elrod-Adams model introduces a strong nonlinearity through the saturation function and its properties. In the next subsubsection, we deal with oblique roughness. Obviously, this case does not fall into Assumption 2.16 which enables us to completely overcome the mentioned difficulties stated in the general case. However, it seems that a change of variables could allow us to recover a structure in which Assumption 2.16 is satisfied. We will see that it is not really the case and that the change of variables will introduce additional terms which are not fully controlled by the homogenization process; nevertheless, it allows us to define, in a rigorous way, two homogenized saturation functions, thus describing anisotropic phenomena on the cavitation. This structure can be considered as an intermediary one between the general case and the microvariables separation case.

30 30 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Oblique roughness Let us consider the mapping defined as: Ê Ê Ü, with ܵ We suppose that the effective gap can be described as follows: ܵ Ó Ò Ü Üµ Ò Ó Ü Assumption 2.26 For a given angle, let be a function such that Ü Üµ Ü Üµ Ü Üµ with Ñ a.e. in ( ). MAX x 2 x 1 MIN Figure 6: Normalized gap with oblique roughness patterns Obviously, heights satisfying Assumption 2.26 (see for instance FIG.6) do not satisfy Assumption 2.16 (except for particular values of ). Let us drop the overscripts (for the sake of simplicity). Now, we say that Ü Ü µ (resp. µ) denotes the original (resp. new) spatial coordinates. So, introducing the vector Ó Ò µ, problem È µ can be described in the coordinates as follows: È Find Ô µ Î Ä µ such that: µöô µö µ µ µ Ö µ Ô Ô µ a.e. in, where µ ܵ and µ, with the following functional spaces: Î Î Ò À µ Ð Ö Ô Ó Ò À µ Ð Ö where Ð (resp. Ö ) denotes the left (resp. right) lateral boundary. Remark 2.27 In the new coordinates, one has µ Ó Î From now on, we denote Ý µ Ý µ and Ý µ Ý µ ( ). Then ݵ Ý µ Ý µ and ݵ Ý µ Ý µ satisfy Assumption 2.16 in the coordinates.

31 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 31 Remark 2.28 The formulation of the lubrication problem in the new coordinates system is equivalent to a generalized Reynolds problem as it happens with an oblique flow direction Ó Ò µ, instead of µ in the classical one. Theorem 2.29 We have the following convergences: (i) There exists Ô À µ such that, up to a subsequence, Ô Ô in À µ and Ô Ô in Ä µ. Moreover Ô Î, and Ô a.e. in. (ii) Ô µ two-scale converges to Ô µ. Moreover, there exists Ô Ýµ Ä À µêµ and a subsequence still denoted such that ÖÔ µ two-scale converges to ÖÔ µ Ö Ý Ô Ýµ. (iii) There exists ݵ Ä µ and a subsequence still denoted such that µ two-scale converges to ݵ. Proof. The result is easily obtained after establishing a priori estimates which do not depend on (see Subsection 2.1). Theorem 2.30 Under Assumption 2.26, one gets the following homogenized problem in the coordinates: È µ Find Ô µ Î Ä µ Ä µ such that: µ ÖÔ µö µ µ Ö µ Î Ô Ô µ µ a.e. in, with the following expressions: and µ µ Moreover problem È µ admits Ô µ as a solution, where µ µ and Ô µ is the two-scale limit of Ô µ (solution of problem È µ).

32 32 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model Proof. We use the same techniques as before, the only modification comes from the presence of an additional term in the right-hand side of the equation. We briefly sketch the main steps of the complete proof: 1st step: Properties of the two-scale limits - Let Ô µ be the two-scale limit of Ô µ (see Theorem 2.29). Then one has: (i) Ô µ in Ä µ, (ii) a.e. in. 2nd step: Macro/microscopic decomposition - Using the classical techniques (previously used in Subsections 2.1 and 2.3), one gets: (i) Macroscopic equation: ÖÔ Ö Ý Ô Ý Ö Ý Ö for every in Î. (ii) Microscopic equation: For a.e., for every À µ. ÖÔ Ö Ý Ô ÖÝ Ý Ö Ý Ý 3rd step: Local problems and macroscopic equation - The local problems µ, Æ µ and Æ µ are identical to the ones defined in Subsection 2.3 (up to the notations adapted to the coordinates). Then, one has: with the following notations: Ï Ý Ï Ý ÖÔ Ö Ï Ý Ï Ý using the notations ( ): Æ Ý and defining the following ratios ( ): Ö Î Æ Ý

33 G. Bayada, S. Martin and C. Vázquez / Two-Scale Homogenization of the Elrod-Adams Model 33 where Ï, and are the solutions of the local problems the analogy with Equations (11), (12) and (13)). µ, Æ µ and Æ µ (consider 4th step: Simplifications due to Assumption Assumption 2.16 in the coordinates (issued from Assumption 2.26) allows us to use the same techniques as in the previous subsubsection to obtain the simplifications on and. Remark 2.31 The earlier formulation is the weak formulation of a generalized Reynoldstype problem including cavitation. The main difference with the initial problem given in the formulation of È µ lies in anistropic effects on the homogenized coefficients, which is a classical result in homogenization theory, but also on the saturation function. Theorem 2.32 [Homogenized exact problem] Under Assumption 2.26, one gets the following homogenized problem in the Ü coordinate: È µ Find Ô µ Î Ä µ Ä µ such that: ÖÔ Ö Ü Î Ô Ô µ µ a.e. in, with the following expressions: ܵ ܵ ܵ ܵ ܵµ Ò Ò Ó Ó Ò Üµ ܵ ܵ ܵ ܵ ܵ ܵ ܵ ܵ ܵ Ò Üµ ܵ Ò Ó, and the following homogenized coefficients ( and ): ܵ ܵ and ܵ ܵ Moreover, problem È µ admits Ô µ as a solution, where ܵ ܵ (32) and Ô µ is the two-scale limit of Ô µ (solution of problem È µ). Proof. Theorem 2.32 is obtained from Theorem 2.30 using the inverse change of coordinates, with ݵ Ü Ýµ. Remark 2.33 Theorem 2.32 implies that we have been able to solve one of the difficulties that raised in the most general case (see Remark 2.11). Indeed there are two saturation functions, but we have proved that they satisfy: ( µ, which was not guaranteed in the general case. In this way, the homogenized problem has a structure that is close to the initial one. But, as in the most general case, we cannot prove a uniqueness result with the methods of Section 1, nor can we numerically solve the problem using algorithms that have been previously mentioned, since we still have two saturation functions. Remark 2.34 Let us recall that, in Theorem 2.10, we wrote the right hand side as, thus defining fake saturation functions (since we were not able to prove that ). In fact, according to Theorem 2.32, should be considered as a combination of, where can be considered as real saturation functions (since they satisfy ).