Reynolds and Weissenberg numbers in viscoelastic flows

Abstract

One of the greatest challenges in performing numerical simulations of viscoelastic flow is how to choose the model parameters in order to be able to achieve quantitative agreement with experimental measurements in benchmark geometries. In the present work, we revisit the original concepts of Reynolds (Re) and Weissenberg (Wi) numbers in order to provide means to adequately choose those parameters. In particular, we highlight the importance of the Oldroyd-B model as a reference for viscoelastic behavior and propose a new methodology for the comparison between general viscoelastic models and the Oldroyd-B model. We identify inconsistencies in the usual approach consensually adopted in the literature. The historical reason for that seems to have its root on the fact that Reynolds and Weissenberg numbers are recognized as expressions and not as concepts. This paradigm allows the choice of parameters of a model as characteristic quantities. We propose here to choose the Oldroyd-B model parameters for the comparison with more complex viscoelastic models so that Re and Wi concepts are maintained. To illustrate this matter, we simulate a flow which is known for capturing an important viscoelastic effect, the transient extrudate swell problem. We tested three viscoelastic models with intermediate levels of complexity, namely PTT, Giesekus, and FENE-P. The traditional and the proposed approaches are compared and the role of the extra parameter, that endows these models with a nonlinear character with respect to the stress, is reinterpreted. It is demonstrated that the nonlinearity introduced by this extra parameter is overestimated in the usual formulation.