On a free boundary problem for the Reynolds equation derived from the Stokes system with Tresca boundary conditions (Q1399386)

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.