hal-00565085, version 2
Rigorous derivation of the thin film approximation with roughness-induced correctors
(2011)
Abstract: 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
- Domain : Mathematics/Analysis of PDEs
- Keywords : Rough boundaries – Reynolds equation – Stokes system – Thin films – Lubrication approximation – Boundary layer
- Comment : 50 pages
- Available versions : v1 (2011-02-11) v2 (2012-07-13)
- hal-00565085, version 2
- http://hal.archives-ouvertes.fr/hal-00565085
- oai:hal.archives-ouvertes.fr:hal-00565085
- From:
- Submitted on: Friday, 13 July 2012 11:59:05
- Updated on: Friday, 13 July 2012 12:28:25



Associated documents
Export