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.
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