  In this article, we derive a rigorous characterization of the boundary-layer
and interior separations in the Taylor-Couette-Poiseuille (TCP) flow. The
results obtained provide a rigorous characterization on how, when and where the
propagating Taylor vortices (PTV) are generated. In particular, contrary to
what is commonly believed, we show that the PTV do not appear after the first
dynamical bifurcation, and they appear only when the Taylor number is further
increased to cross another critical value so that a structural bifurcation
occurs. This structural bifurcation corresponds to the boundary-layer and
interior separations of the flow structure in the physical space.
