  In the present work, we study the flow of thixotropic yield stress fluids
between two concentric cylinders. In order to take into account the thixotropy,
the constitutive relation uses a structural parameter which is driven by a
kinetic equation. Here, the Houska's model is considered. Depending on the
breakdown rate of the structural parameter, localization or shear-banding are
observed. We show that for fragile structures, a shear-banding flow may be
observed although for stronger structures, only localisation of the flow is
observed such as in Bingham fluids. Physical explanations of the shear-banding
discussed by several authors in the literature highlight that the shear-banding
may be associated with a discontinuity into the structure of the material and a
non-monotonic evolution of the stress according to the constitutive relation
with the strain rate. Solving numerically the flow, we show that such a
rheological model based on the existence of a structural parameter is able to
predict shear-banding. Moreover, the consequences of the thixotropy on the
linear stability of the azimuthal flow is studied in a large range of
parameters. Although the thixotropy allows shear banding for the base flow, it
does not modify fundamentaly the stability of the Couette flow compared to a
simple yield stress fluid. The apparent shear-thinning behaviour depends on the
thixotropic parameters of the fluid and the results about the onset of the
Taylor vortices in shear-thinning fluids are retrieved. Nevertheless, the
shear-banding modifies the stratification of the viscosity in the flowing zone
such that the critical conditions are mainly driven by the width of the flowing
region.
