Elastic and Viscous Effects in Viscoelastic Flows: Elucidating the Distinct Roles of the Deborah and Weissenberg Numbers
The interpretation of the parameters appearing in constitutive models for viscoelastic fluids is essential for analyzing theoretical predictions and understanding the origin of phenomena observed in experiments. In this work, we examine the physical significance of the Deborah ($De$) and Weissenberg ($Wi$) numbers, along with other key parameters commonly used in these models. The central objective is to clarify the extent to which these dimensionless groups effectively characterise the competition between elastic and viscous effects in complex flows. While these parameters are ubiquitous in theoretical and experimental research, their interpretation is often context-dependent and prone to ambiguity. To address this, we analyse two representative scenarios: an analytical solution for unsteady planar flow and a numerical simulation of viscoelastic flow between rotating coaxial cylinders, governed by the Oldroyd-B constitutive equations. Our findings elucidate the distinct roles of these dimensionless numbers, offering guidelines for their rigorous interpretation in both analytical and numerical studies.