Asymptotic thin films. Viscoelastic models.

The purpose of this work is to provide a rigourous justification of the validity of a model of a thin film flow for viscoelastic fluids described by a law of the Oldroyd type. In many mechanical works, the authors obtained in a heuristic way various models of viscoelastic fluids in thin films, for the generalized Maxwell model [4] or for Phan Thien-Tanner laws [1]. In this work, a mathematical approach is developed for the Oldroyd-B model. Viscoelastic fluids are non-newtonian fluids whose constitutive equation (relation between the stress tensor and the shearing rate) is a differential law. By coupling this law with the Navier-Stokes equations, one obtains the following macroscopic model: ∂tU + U · ∇U − η(1 − r)∆U +∇p − div σ = 0, divU = 0, λ (∂tσ + U · ∇σ + g(∇U, σ)) + σ = 2rηD(U). This model couples equations on the velocity U of the fluid, its pressure p and the stress tensor σ. The existence of a global solution to this problem has been proved in [3]. Supposing the domain to be thin (one dimension is small with respect to the two others), it was shown in [2] that one obtains formally a system coupling only the velocity U ∗ = (u∗, w∗) ∈ R2 ×R and the pressure p∗, for which it is possible to show the existence of a weak solution: − η(1 − r)∂