In this paper, we present an analytical study of the stability of a flexible pipe-conveying non-uniform viscous flow in order to characterize the fatigue situation which induces damage. Based on the fluid–structure interaction mechanisms in the pipe-conveying fluid, we provide physical mechanisms responsible for the emergence of helical-type vortices in transient pipe flows. Additionally, we establish the Lange–Newell criterion for the appearance of flow patterns in the pipe. To achieve these goals, the nonlinear generic models are first transformed into a two-dimensional coupled complex Ginzburg–Landau equation. Thereafter, this equation is explored to perform linear stability analysis, and some modulational instability properties are derived and found to be modified by the change in the Reynolds number, which is set as the control parameter. A bifurcation analysis is performed analytically and numerically and based on these, we find that the system undergoes Hopf or Bautin bifurcations. As a result, a phase portrait-like helical-type vortices of the velocity, pressure, and pipe displacement are obtained. Moreover, frequency analysis is implemented to show the influence of parameters on the power spectral density curve. The results obtained in this work find applications in aeronautics for sizing reactors, in the water processing industry, in the oil industry, in hydraulics for sizing flexible pipes, and in heat exchangers in the nuclear industry.
Kaptue et al. (2026) studied this question.