From the Phan–Thien–Tanner/Oldroyd-B Non-Newtonian Model to the Double Shear Thining Rabinowisch Thin Film Model

In this paper, an asymptotic expansion is used to derive a description of Phan–Tien– Tanner (PTT)/Oldroyd-B flows in the thin film situation without the classical “upper convective maxwell”(UCM) assumption. We begin with a short presentation of the Phan–Thien–Tanner/Oldroyd-B models, which introduce viscoelastic effects in a solute–solvent mixture. The three-dimensional flow is described using five parameters, namely the Deborah number (De) (or the relaxation parameter $λ$⁠), the viscosity ratio $r$⁠, the bulk fluid viscosity $η$⁠, the material slip parameter $a$ related to the “convected derivative” and an elongation number $κ$⁠. 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 $κ$ 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.