Résumé
We report on the response of semi-dilute shear-banding wormlike micelles to time-dependent flow protocols such as step stress and shear startup. We use global rheology coupled to direct optical visualisations in Taylor-Couette flow to provide a detailed 2D description of the spatio-temporal dynamics of the shear banding flow under imposed stress. We compare these dynamics to their extensively studied counterpart under imposed shear rate. We show that, for a given asymptotic state, the onset of banding is delayed and the transients are longer when the stress is used as control parameter. The way the banding structure is built up strongly depends on the control parameter. In the two cases, secondary flows develop on top of the shear-banding instability: after shear startup, the interface between bands is formed in the bulk of the system and migrates inwards to its stationary position in the gap of the Taylor-Couette device, before being destabilized due to elastic instability; under creep flow, the interface between bands forms at the inner wall and becomes wavy while it is still migrating outwards to its steady position, leading to permanent rearrangements of the flow structure until steady state is reached, potentially explaining longer transients. We also outline the possible mechanical signature of the onset of shear banding in terms of shape of the bulk mechanical signals, which is the subject of the second part of this work [Briole et al., submitted].