Abstract The flow models of a Taylor-Couette-Poiseuille flow with a wavy interior surface have been studied at low axial Reynolds numbers (Reax less than or equal to 32) and Taylor numbers (Ta less than or equal to 606). Two types of surfaces were operated; smooth and wavy. The surface of the wavy cylinder has 18 identical axial waves, which are arranged periodically. The Taylor-Couette system is characterized by an average inner and outer cylinder radius ratio of 0.9, an average gap between the two concentric cylinders of 10mm and an aspect ratio equal to 45, corresponding to the dimensionless height of these cylinders. The spatiotemporal flow representation was provided quantitatively and qualitatively by exploiting flow visualizations and simultaneous measurements of the components of the wall shear rate. An axial flow imposed on the base flow not only modifies the conditions for the appearance of instabilities, but also the vortex structure of vortices. It can have a stabilizing effect for a regime flow and a destabilizing effect for another, i.e., the transition to Taylor vortices occurred at higher or lower Taylor numbers. In such destabilizing cases, the vortices are either folded or stretched or overlapped or completely destructed. As a function of the axial Reynolds and Taylor numbers, helices appear, develop and wind in the same direction as the base flow or in the opposite direction. It has been observed that the shear stress of the wall strongly depends on the nature of the surface in contact with the flow, on Ta and Re.