  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.
