hal-00565085, version 1
Rigorous derivation of the thin film approximation with roughness-induced correctors
(2011)
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.
- 1 :
- CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II
- 2 :
- CNRS : UMR8628 – Université Paris XI - Paris Sud
- Domaine : Mathématiques/Equations aux dérivées partielles
- Mots-clés : Rough boundaries – Reynolds equation – Stokes system – Thin films – Lubrication approximation – Boundary layer
- Commentaire : 50 pages
- Versions disponibles : v1 (11-02-2011) v2 (13-07-2012)
- hal-00565085, version 1
- http://hal.archives-ouvertes.fr/hal-00565085
- oai:hal.archives-ouvertes.fr:hal-00565085
- Contributeur :
- Soumis le : Jeudi 10 Février 2011, 22:45:03
- Dernière modification le : Vendredi 11 Février 2011, 08:27:51



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