  The Taylor-Couette flow of a dilute micellar system known to generate
shear-induced structures is investigated through simultaneous rheometry and
ultrasonic imaging. We show that flow instabilities must be taken into account
since both the Reynolds number and the Weissenberg number may be large. Before
nucleation of shear-induced structures, the flow can be inertially unstable,
but once shear-induced structures are nucleated the kinematics of the flow
become chaotic, in a pattern reminiscent of the inertio-elastic turbulence
known in dilute polymer solutions. We outline a general framework for the
interplay between flow instabilities and flow-induced structures.
