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 / 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 De the Deborah number (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. 1 Nomenclature H Order of magnitude of the gap L Order of magnitude of the bearing length U Order of magnitude of the shear velocity L1, L2 Dimensions of the contact area H (h) Gap (dimensionless) ε = H /L Film thickness ratio Re Reynolds number De (D̃e = De/ε) Deborah number (rescaled) κ (K = κ/r, K̃ = K/ε) Elongation number (rescaled) b, c, n, G Parameters in Carreau-Yosida law a Material slip parameter in the time derivative r Ratio of viscosities or retardation number λ Relaxation time ρ Density δ Convergence parameters in the algorithm δx1 , δx2 Discretization parameters s Dimensionless velocity at z = 0 in the direction x1 t⋆ (t) Time (dimensionless) p⋆ (p) Pressure (dimensionless) q0 Input mass flux value U⋆ = (u1,u ⋆ 2,w ⋆) (U = (u1,u2,w)) Velocity (dimensionless) x = (x1,x ⋆ 2), (x = (x1,x2)) Coordinates on the shearing surface (dimensionless) z (z) Orthogonal coordinate (dimensionless) D(U) = 2 ( ∇U+ T(∇U) ) Symmetric parts of the velocity gradient W (U) = 2 ( ∇U− T(∇U) ) Skew-symmetric parts of the velocity gradient γ̇ (α̇, β̇) Shear rate (dimensionless 1D -for long devicesor 2D) η1, η2, ηel., ηvisc., ηeff, η Viscosities σ⋆, σel., σ ⋆ visc. (σ, σel., σvisc.) Stress tensor (dimensionless) τ Shear stress ω Contact area