From the Phan-Thien Tanner / Oldroyd-B non-Newtonian model to the double shear thining Rabinowisch thin film model

International audienceIn this paper, an asymptotic expansion is used to derive a description of Phan-Tien Tanner~/ Oldroyd-B flows in the thin film situation without the classical "Upper Convective Maxwell'' assumption. We begin with a short presentation of the Phan-Thien Tanner~/ Oldroyd-B models which introduce viscoelastic effects in a solute-solvant mixture. The three dimensional flow is described using five parameters, namely the Deborah number (or the relaxation parameter), the viscosity ratio, the bulk fluid viscosity, the material slip parameter related to the "convected derivative'' and an elongation number. Then we focus on the thin film assumption and the related asymptotic analysis that allows us to derive a reduced model. A perturbation procedure for "not too small'' values of the elongation number allows us to obtain further results such as an asymptotic "effective viscosity / shear rate'' law which appears to be a perturbation of the double Rabinowisch model whose parameters are completely defined by those of the original three dimensional model. And last a numerical procedure is proposed based on a penalized Uzawa method, to compute the corresponding solution. This algorithm can also be used for any generalized double Newtonian shear thinning Carreau law