Two-scale homogenization of a hydrodynamic Elrod-Adams model

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 r=r = 10 3 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a long time to describe the behaviour of a viscous flow between two close surfaces in relative motion (see [37, 38] for historical references).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 r h3rp = x1 h ; where p is the pressure distribution, and h 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is defined as the rupture of the continuous film due to the formation of air bubbles and makes the Reynolds equation no longer valid in the cavitation area.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 (P ) in the next section). From a mathematical point of view, the problem can be simplified using a penalized Reynolds equation with cavitation (see problem (P ) 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 "= 1, " 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 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. 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 =℄0; 2 [ ℄0; 1[ and we suppose that the following assumptions are satisfied: Assumption 1.1 h 2 C1( ) is 2 x1 periodic and satisfies 9 h0; h1; 8 x 2 ; 0 < h0 h(x) h1: Assumption 1.2 pa is a Lipschitz continuous non-negative function, 2 periodic. Now let us introduce the Elrod-Adams model taking into account cavitation phenomena. Thus we introduce an exact problem and a penalized problem. (i) Exact Reynolds problem The strong formulation of the problem is given by the following set of equations: 8<: r h3(x)rp(x) = x1 (x) h(x) ; x 2 p(x) 0; p(x) (1 (x)) = 0; 0 (x) 1; x 2 with the following boundary conditions: p = 0 on 0 and p = pa on a, (Dirichlet conditions) h h3 p x1 and p are 2 x1 periodic, (periodic conditions) where (x) is the normalized height of fluid between the two surfaces. The boundaries 0 and a 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