Rigorous Derivation of the Thin Film Approximation with Roughness-Induced Correctors

We derive the thin film approximation including roughness-induced correctors. This corresponds to the description of a confined Stokes flow whose thickness is of order ε (designed to be small) ; but we also take into account the roughness patterns of the boundary that are described at order ε, leading to a perturbation of the classical Reynolds approximation. The asymptotic expansion leading to the description of the scale effects is rigorously derived, through a sequence of Reynolds-type problems and Stokes-type (boundary layer) problems. Well-posedness of the related problems and estimates in suitable functional spaces are proved, at any order of the expansion. In particular, we show that the micro-/macroscale coupling effects may be analysed as the consequence of two features: the interaction between the macroscopic scale (order 1) of the flow and the microscopic scale (order ε of the thin film) is perturbed by the interaction with a microscopic scale of order ε related to the roughness patterns (as expected through the classical Reynolds approximation) ; moreover, the converging-diverging profile of the confined flow, which is typical in lubrication theory (note that the case of a constant cross-section channel has no interest) provides additional micro-macro-scales coupling effects. This work was supported by the ANR project ANR-08-JCJC-0104-01 : RUGO (Analyse et calcul des effets de rugosités sur les écoulements). The authors are very grateful to David Gérard-Varet for fruitful discussions on the subject of this paper. Université Blaise Pascal, Clermont-Ferrand II, Laboratoire de Mathématiques CNRSUMR 6620, Campus des Cézeaux, F-63177 Aubière cedex, France. [email protected] Université Paris-Sud XI, Laboratoire de Mathématiques CNRS-UMR 8628, Faculté des Sciences d’Orsay, Bât. 425, F-91405 Orsay cedex, France. [email protected] 1 ha l-0 05 65 08 5, v er si on 1 10 F eb 2 01 1