Rigorous derivation of the thin film approximation with roughness-induced correctors
Résumé
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~$\eps$ (designed to be small)~; but we also take into account the roughness patterns of the boundary that are described at order~$\eps^2$, 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-/macro-scale 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~$\eps$ of the thin film) is perturbed by the interaction with a microscopic scale of order~$\eps^2$ 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.
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