The pressure-driven displacement flow of a less viscous fluid initially occupying an axisymmetric pipe by a highly viscous fluid is investigated. The fluids consist of two species diffusing at different rates. The fluids are assumed to be Newtonian, incompressible with the same density, but of different viscosity modelled as an exponential function of the concentration of both the species. A parametric study investigating the effects of diffusivity ratio, log-mobility ratios of the slower and faster diffusing species and Reynolds number on the flow dynamics is conducted. Our results demonstrate the presence of instability patterns due to double-diffusive effect in situations when a less viscous fluid displaces a highly viscous fluid. These instabilities are qualitatively different from those observed in planar channel. The intensity of these instabilities increases with increasing the values of diffusivity ratio. It is demonstrated that a highly viscous stenosis region is created near the entrance of the pipe due to double-diffusive effect, providing a favourable condition to start the instability. In addition to this, because of double-diffusive effect locally at some portion of the pipe, the less viscous fluid becomes the displacing fluid, which promotes the development of instability. © 2015 Elsevier Masson SAS.