Elasto-Inertial Instability in Torsional Flows of Shear-Thinning Viscoelastic Fluids

It is well known that inertia-free shearing flows of a viscoelastic fluid with curved streamlines, such as the torsional flow between a rotating cone and plate, or the flow in a Taylor-Couette geometry, can become unstable to a three-dimensional time-dependent instability at conditions exceeding a critical Weissenberg (Wi) number. However, the combined effects of fluid elasticity, shear thinning, and finite inertia (as quantified by the Reynolds number Re) on the onset of elasto-inertial instabilities are not fully understood. Using a set of cone-plate geometries, we experimentally explore the entire Wi - Re phase space for a series of rate-dependent viscoelastic fluids (quantified using a shear thinning parameter $\beta_P$). We tune $\beta_P$ by varying the polymer concentration in solutions. This progressively reduces shear-thinning but leads to finite inertial effects before the onset of elastic instability, thus naturally resulting in elasto-inertial coupling. Transient rheometric measurements and flow visualization experiments allow us to investigate the effects of flow geometry and document the combined effects of varying Wi, Re, and $\beta_P$ on the emergence of secondary motions at the onset of instability. The resulting critical state diagram quantitatively depicts the competition between the stabilizing effects of shear thinning and the destabilizing effects of inertia. We extend the curved streamline instability criterion of Pakdel and McKinley 1996 for the onset of purely elastic instability in curvilinear geometries by using scaling arguments to incorporate shear thinning and finite inertial effects. The augmented condition facilitates predictions of the onset of instability over a broader range of flow conditions, thus bridging the gap between purely elastic and elasto-inertial curved streamline instabilities.