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

We analyze transient dynamics during the 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, nonstretching Rolie-Poly, and Giesekus models. We explore the dynamics of the 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 quasisteady 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 coevolve 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 the shear start-up within the ambit of a single model-independent criterion. Our work also suggests that the strong transients during the shear start-up seen in earlier work could be sensitive to the small solvent viscosity values considered in the absence of otherwise negligible terms such as fluid inertia.