Boundary-value-problem examination of the stability of a symmetrical rotor partially filled with a viscous incompressible fluid

Abstract

The instabilities in a rotor system partially filled with a fluid can have an exponentially increasing amplitude, and this can cause catastrophic damage. Numerous theoretical models have been proposed and numerous experiments have been conducted to investigate the mechanisms of this phenomenon. However, the explanation of the existence of the first unstable region induced by a viscous incompressible fluid is unclear and only one solving method, a standard finite difference procedure, was proposed in 1991 for solving the instabilities in a system containing a symmetric rotor partially filled with a viscous incompressible fluid. To better understand the mechanisms of the instability induced by the viscous fluid, based on the linearized two-dimensional Navier--Stokes equations, this system's differential equations are transferred to solve the characteristic equations with boundary conditions. A \textsc{Matlab} boundary value problem (BVP) solver bvp5c proposed in 2008 is an efficient tool to solve this problem by uncoupling the boundary conditions with unknown initial guess. Applying this approach to a rotor system allows the instability regions to be obtained. In this study, first, the radial and tangential velocities and pressure fluctuations along the radial direction of a disk filled with fluid were examined. Then, parametric analysis of the effect of the Reynolds number $\textit{Re}_{cr}$, filling ratio $H$, damping ratio $C$, and mass ratio $m$ on the system's stability was conducted. Using this calculation method allowed the first exploration of some new laws regarding the instabilities.