Affiliations: [a] Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, Bat. Braconnier, 43, blvd du 11 novembre 1918, F-69622 Villeurbanne Cedex, France. E-mail: [email protected] | [b] Université de Lyon, Insa de Lyon, CNRS, UMR 5208, Institut Camille Jordan, Bat Léonard Vinci, 20 Av. A. Einstein, F-69621 Villeurbanne Cedex, France. E-mail: [email protected] | [c] Matemática Aplicada a las T.I.C., ETSI Sistemas Infomáticos, Universidad Politécnica de Madrid, 28031 Madrid, Spain
Abstract: In this article we study a lubricated system consisting on a slider moving over a smooth surface and a known external force (the load) applied upon the slider. The slider moves at constant velocity and close proximity to the surface and the gap is filled by an incompressible fluid (the lubricant). At the equilibrium, the position of the slider presents one degree of freedom to be determined by the balance of forces acting on the system: the load and the total force exerted by the pressure of the lubricant. The pressure distribution is described by a variational inequality of elliptic type known as Swift–Stieber model and based on Reynolds equation. The distance h between the surfaces in a two dimensional domain Ω is given by
hη(x1,x2,y)=h0(x1,x2)+h1(y)+η,(x1,x2)∈Ω,y∈[0,1]
where h0(x1,x2)∼|x1|α for α>0 and h1(y)∼|y−y0|β for y being the homogenization variable. The main result of the article quantify the influence of the roughness in the load capacity of the mechanism in the following way:
If α<3γfor 0<γ⩽2α<min{1γ−2,3γ}for γ>2
then, the mechanism presents finite load capacity, i.e. limη→0∫Ωpη<∞. Infinite load capacity is obtained for γ>1 and α>2/(γ−1). A one dimensional particular case is given for γ>3/2 with infinite load capacity.
Keywords: Lubrication, Reynolds variational inequality, homogenization, inverse problem
