On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions
type: Item
instance of: scholarly article
title: On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions
zbMATH Open document ID: 1025.35022
DOI: 10.1016/S0022-247X(03)00140-9
author: Guy Bayada
author: Mahdi Boukrouche
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: When considering the flow of a fluid around a solid body, one has to describe the appropriate boundary conditions on the fluid-solid interface, specially when a slip condition is introduced. The authors intend to derive such boundary conditions using an asymptotic analysis approach. They consider a thin film fluid occupying a domain \(\Omega ^{\varepsilon }=\omega \times ( 0,\varepsilon h( .)) \) with varying thinness. On the upper part of the domain the velocity is imposed to be equal to 0. On the lateral boundary the velocity satisfies a zero flux condition and with a vanishing third component. On the lower part of the domain the velocity must satisfy \(u^{\varepsilon }\cdot n=0\) and the tangential velocity verifies a Tresca friction law involving a friction coefficient \(k^{\varepsilon }\). The fluid flow obeys the incompressible Stokes law. Assuming that the friction coefficient \(k^{\varepsilon }\) is a nonnegative function in \(L^{\infty }( \omega) \) the authors first prove an existence and uniqueness result for this Stokes problem, considering the variational formulation of the problem. The main part of the work is devoted to the description of the asymptotic behaviour of the solution when the thinness parameter \(\varepsilon \) of the film goes to 0. First introducing a change of scale the authors work in a fixed domain and imposing some growth conditions on the body forces and on the friction coefficient they get estimates on the solution which allow to take the limit and to derive a limit slip law of Reynolds type. The authors conclude their work proving a uniqueness result for this limit problem which can be described through variational inequalities.
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 76A20
Mathematics Subject Classification ID: 76D08
Mathematics Subject Classification ID: 76M45
zbMATH DE Number: 1956912
zbMATH Keywords: asymptotic approach
zbMATH Keywords: Reynolds equation
zbMATH Keywords: Tresca fluid-slide condition
zbMATH Keywords: lubrication
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: variational inequalities
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1956912