  In the past twenty years, shear-banding flows have been probed by various
techniques, such as rheometry, velocimetry and flow birefringence. In micellar
solutions, many of the data collected exhibit unexplained spatio-temporal
fluctuations. Recently, it has been suggested that those fluctuations originate
from a purely elastic instability of the flow. In cylindrical Couette geometry,
the instability is reminiscent of the Taylor-like instability observed in
viscoelastic polymer solutions. In this letter, we describe how the criterion
for purely elastic Taylor-Couette instability should be adapted to
shear-banding flows. We derive three categories of shear-banding flows with
curved streamlines, depending on their stability.
