A Generalized Approach to the Linear Stability of Viscoelastic Shear-Flows

Our talk revisits the linear stability theory of viscoelastic shear-flows, based upon a constitutive equation of fading-memory-type. The particular model was introduced by Kenneth Walters through the integration of classical rate-type fluids in a convected frame. Despite their broad applicability for various materials, their utilization in shear flow stability analysis has been insufficiently explored. Initial findings by Tackels and Crochet revealed promising agreement for small Weissenberg numbers but raised questions for larger values. The talk provides a concise formulation of these foundational results in terms of a displacement field, a well-known concept in the theory of linear elasticity, inheriting several advantages. Firstly, the analysis of Tackels and Crochet is extended to diverse material models, including those of rate-type without exact solutions in terms of convected integration. Furthermore, no assumptions on the kinematics of the base flow had to be imposed and the study is readily extended to the weakly non-linear stability analysis.

Finally, some strides are made to study the linear but non-modal stability of linear shear-flows, based upon a kelvin-type ansatz.