Onset of transient shear banding in viscoelastic shear start-up flows: Implications from linearized dynamics

TL;DR

This paper investigates the onset of transient shear banding during shear start-up in viscoelastic fluids using linearized stability analysis and numerical methods, revealing complex dynamics that challenge simple criteria for instability.

Contribution

It introduces a rigorous numerical approach to analyze transient growth in shear start-up flows and clarifies the non-universality of stress overshoot as an indicator of shear banding onset.

Findings

No universal link between stress overshoot and shear banding.

Transient growth can occur without eigenvalue instability.

Inertia and solvent viscosity influence transient shear banding.

Abstract

We analyze transient dynamics during shear start-up in viscoelastic flows between two parallel plates, with a specific focus on the signatures for the onset of transient shear banding using the Johnson-Segalman, non-stretching Rolie-Poly and Giesekus models. We explore the dynamics of shear start-up in monotonic regions of the constitutive curves using two different methodologies: (i) the oft-used `frozen-time' linear stability (eigenvalue) analysis, wherein we examine whether infinitesimal perturbations imposed on instantaneous stress components (treated as quasi steady states) exhibit exponential growth, and (ii) the more mathematically rigorous fundamental-matrix approach that characterizes the transient growth via a numerical solution of the time-dependent linearized governing equations, wherein the linearized perturbations co-evolve with the start-up shear flow. Our results reinforce the hitherto understated point that there is no universal connection between the overshoot and subsequent decay of shear stress in the base state and the unstable eigenvalues obtained from the frozen-time stability analysis. It may therefore be difficult to subsume the occurrence of transient shear banding during shear start-up within the ambit of a single model-independent criterion. Our work also suggests that the strong transients during shear start-up seen in earlier work could well be a consequence of consideration of the limit of small solvent viscosity in the absence of otherwise negligible terms such as fluid inertia.