ABSTRACT The present study investigates the dynamic behavior of polymer nanocomposite micropipes conveying a viscous fluid. The fundamental equation governing the motion of such fluid‐conveying micropipes is formulated using Hamilton's principle within the frameworks of Euler–Bernoulli beam theory and modified couple stress theory. Using the Ablowitz–Ramani–Segur (ARS) algorithm, the governing equation is proved to be non‐integrable in the Painlevé sense. A Galilean transformation is subsequently applied to convert the governing equation into a non‐conservative two‐dimensional dynamical system. The qualitative theory of planar systems is employed to analyze bifurcation behavior and construct phase portraits in the parameter plane, where the parameters depend on both the material properties and the characteristics of the viscous fluid. By applying Bendixson's criterion, the non‐existence of periodic phase orbits is established, implying the absence of periodic axial and transverse displacement waves—except when the wave velocity attains a critical value determined by the fluid's properties. At this critical velocity, the system becomes conservative and exhibits a Hamiltonian structure, enabling a detailed bifurcation analysis. Several new analytical solutions for axial and transverse displacement waves are obtained and classified into super‐periodic, periodic, solitary, and kink (or anti‐kink) types. The degeneracy of these solutions is investigated through their phase orbits, which confirm consistency and physical validity of the waveforms. Both two‐ and three‐dimensional visualizations are presented for various combinations of material and fluid parameters. In addition, the effects of graphene weight fraction, material length scale, Knudsen number, and mean flow velocity are analyzed. These factors are shown to significantly influence the amplitude and width of the axial and transverse waves.
A. A. Elmandouh (Mon,) studied this question.