Nonlinear dynamics of a vertical pipe subjected to a two-phase, solid-liquid internal flow

In this paper, a numerical study is carried out on a simply supported pipe model which conveys a two-phase, solid-liquid flow. First, a system of nonlinear, partial differential equations of dimensionless, two-phase, solid-liquid flow pipeline is established. Then, the above mentioned partial differential equation system is discretized by the Galerkin method of finite order to obtain a system of ordinary differential equations. Finally, the natural frequency characteristics of the corresponding linear system are solved to approximate the stability range of the nonlinear system, the structural response characteristics of which are solved by the Newton-Raphson method. The dimensionless aspect ratio, the dimensionless liquid-phase volume, and the dimensionless gravity coefficient are chosen to analyze the impact on the critical velocity of buckling and structural nonlinear response characteristic in this research. The research results indicate that a change to the above parameters will cause the critical speed to change in a linear or non-linear form. Moreover, at subcritical speeds, the vibrations are damped; at critical speeds, steady-state vibrations with extremely small amplitudes with a single frequency are seen; and at supercritical speeds, the pipeline model will produce buckling in different directions according to initial values.