Influence of steady viscous forces on the non-linear behaviour of cantilevered circular cylindrical shells conveying fluid

In this study, the effect of steady viscous forces (skin friction and pressurization) on the non-linear behaviour and stability of cantilevered shells conveying fluid is investigated for the first time. These forces are obtained by using the time-mean Navier-Stokes equations and are modelled as initial loadings on the shell, which are in a membrane-state of equilibrium with in-plane stresses. The unsteady fluid-dynamic forces, associated to shell motions, act as additional loadings on this pre-stressed configuration; they are modelled by means of potential flow theory and obtained by employing the Fourier transform technique. The problem is formulated using the extended Hamilton's principle in which the shell model is geometrically non-linear and based on Flügge's thin shell assumptions. This model includes non-linear terms of mid-surface stretching and the non-linear terms of curvature changes and twist, as well. The displacement components of the shell are expanded by using trigonometric functions for the circumferential direction and the cantilevered beam eigenfunctions for the longitudinal direction. Axisymmetric modes are successfully incorporated into the solution expansion based on a physical approximation. The system is discretized and the resulting coupled non-linear ODEs are integrated numerically, and bifurcation analyses are performed using the AUTO program. Results show that the steady viscous effects diminish the critical flow velocity of flutter and extend the range of flow velocity over which limit cycle responses are stable. On the other hand, the non-linear terms of curvature changes and twist have very little effect on the dynamics. The system exhibits rich post-critical dynamical behaviour and follows a quasiperiodic route to chaos.