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hal-00565085, version 2

Rigorous derivation of the thin film approximation with roughness-induced correctors

Laurent Chupin () 1, Sébastien Martin () 2

(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 :  Laboratoire de Mathématiques
  • CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II
  • 2 :  Laboratoire de Mathématiques d'Orsay (LM-Orsay)
  • 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 2
  • oai:hal.archives-ouvertes.fr:hal-00565085
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  • Soumis le : Vendredi 13 Juillet 2012, 11:59:05
  • Dernière modification le : Vendredi 13 Juillet 2012, 12:28:25