Improved convergence based on linear and non-linear transformations at low and high Weissenberg asymptotic analysis

Abstract

We investigate the accuracy and efficiency of linear and non-linear transformations on the truncated series solutions obtained by asymptotic methods for simple shear and uniaxial elongational viscoelastic flows. The Upper Convected Maxwell, Oldroyd-B, exponential and linear Phan-Thien and Tanner, Giesekus, and FENE-P differential constitutive models are used. The major dimensionless quantity of interest is the Weissenberg number (Wi) in terms of which the unknown dependent flow variables are expanded suitably in series form. The standard regular perturbation method is performed at the Newtonian limit (zero Weissenberg number), while regular or singular perturbation techniques are required to perform the analysis at the purely elastic limit (infinite Weissenberg number). It is shown that the asymptotic approximations at low Wi converge very slowly and have limited radius of convergence, while at high Wi the asymptotic approximations are very accurate even for Wi close to unity. Further improved convergence is achieved by applying a variety of techniques on the original asymptotic solutions, such as Euler's linear transform, Shanks’ non-linear transform, and Padé-type approximants. It is found that the accuracy and the radius of convergence of the transformed solutions increase substantially compared to the original ones. By combining the transformed solutions from both limits, excellent agreement with the exact solutions is demonstrated over the entire range of the Weissenberg number, from zero to infinity, except for a very narrow region close to Wi= 1/2 under uniaxial elongational flow. The results show that, contrary to intuition, viscoelastic phenomena at Wi=O(1) can be predicted accurately using non-linear analysis.