Dynamics of horizontal pipes conveying two phase flow with nonlinear boundary conditions

The occurrence of pipe conveying two phase flow is common in many engineering systems such as nuclear reactors, refrigeration systems, pipes conveying crude oil and gas. In practice, the model systems are subjected to complex nonlinear boundary conditions. Unfortunately during modelling, these complex boundary conditions are conveniently replaced with simplified ones, thereby ignoring nonlinearities. In this work, horizontal pipes conveying two phase flows subjected to nonlinear boundary condition is studied for the first time. The process described by a nonlinear partial differential equation is discretized to a set of nonlinearly coupled ordinary differential equations (ODEs) by the Galerkin method. The linear analysis was investigated for three distinct cases, for which detailed investigation was further conducted for a total of five subcases. The harmonic balance method (HBM) is used to investigate the steady-state responses of the slightly curved pipes. The simulations yield the amplitude–frequency responses of the system. These outcomes are validated by the numerical solutions via Runge–Kutta method and finite difference method (FDM). Exploring special cases of the nonlinear boundary conditions suggest that nonlinear boundary conditions can be sub divided into three cases, and these cases are characterized in their linear and steady states responses. The three cases have their responses in the linear case but steady state is only possible in one of them. Furthermore, increase in the initial curvature of the pipe and vapour quality of the two phase flows have increasing and reducing effect on the resonance frequency of the pipe. Therefore, this work further shed more lights on the vibration of slightly curved pipes with nonlinear boundary conditions.