  For the first time, the development of the nonlinear geometrically exact
governing equations and corresponding boundary conditions of hanging
cantilevered flexible pipes conveying fluid in the framework of the quaternion
system is undertaken. Additionally, the linear model is derived from the
nonlinear one for the stability analysis. The linear and nonlinear mathematical
formulations based on the geometrically exact rotation-based model are also
extracted from the current model. The obtained integro-partial
differential-algebraic equations are converted to a set of ordinary
differential algebraic equations via the Galerkin discretization technique, and
the resulting equations are numerically solved to determine the self-excited
oscillation behavior of the system in the post-flutter region. The critical
flow velocities, time traces, bifurcation diagrams, phase planes, and deformed
configurations are compared with those reported based on the geometrically
exact rotation-based model. The comparative studies divulge that the present
model can successfully capture the stability and dynamic characteristics of the
system. An interesting feature of the current ordinary differential-algebraic
equations is that their coefficients are time-independent, unlike those of
geometrically exact rotation-based equations in the Galerkin form. The results
show this feature leads to a remarkable decrease in the computational cost,
although the number of equations increases and the nonlinearity becomes
stronger. The challenging issues with the numerical solution utilized in this
study are also discussed.
