Improved convergence based on two-point Padé approximants: Simple shear, uniaxial elongation, and flow past a sphere

We investigate the performance of two-point Padé approximants on the truncated series solutions for the shear and elongational viscosities (obtained directly from the exact solutions or by asymptotic methods) for viscoelastic flows modeled with the Giesekus constitutive equation. Τhe major dimensionless groups are the Weissenberg number (Wi), the Giesekus mobility parameter (α), and the ratio of the polymer viscosity to the total viscosity of the fluid (η_p) in terms of which the exact analytical solutions for the shear and elongational viscosities are found. The exact solutions are used to derive truncated series expansions in terms of the Weissenberg number at the Newtonian limit (zero Wi) and in terms of the inverse of the Weissenberg number at the purely elastic limit (infinite Wi). These series are then used to construct low- and high-order two-point diagonal Padé approximants. It is found that both the low- and high-order formulas predict the right trends over the entire range of Wi, from the Newtonian limit to the pure elastic limit. As expected, the high-order formula is much more accurate than the low-order formula. The results show that viscoelastic phenomena, such as shear thinning and extensional thickening, at any value of the Weissenberg number can be predicted analytically with acceptable accuracy using the proposed two-point Padé non-linear analysis. Finally, a fourth-order formula in terms of the Weissenberg number (or in terms of the Deborah number) for the drag force on a rigid sphere that translates with constant speed in a Giesekus fluid is also provided along with a discussion regarding the missing information from the literature, that is, required to perform the proposed non-linear analysis.