自由下降液膜流动的长波方程的稳定性分析

采用行波方法将长波方程变换为动力系统的自治方程，以分析在渐近稳定状态下自由下降液膜流动的稳定性。通过应用Maple数学软件对自治方程分析求解，获得了以雷诺数，韦伯数和波速为控制参数的特征方程。根据临界点附近的特征值的复数情况，发现在临界点附近不存在霍普夫分叉现象。非线性和线性稳定性分析均表明短波和长波在液膜流动中分别起着不稳定作用和稳定作用。分析表明，行波方法是一种具有简化偏生分方程分析的有效工具，长波方程不适于描述雷诺数较大的液膜流动。

Stability Analysis of Long Wave Equation for the Free Falling Flow of Liquid Film

Autonomous equations of dynamical system are transformed from the long wave equation by traveling wave method to study the stability of free falling flow of liquid films. After solving the autonomous equations by Maple software, equations of eigenvalue,containing Reynolds number, Webber number and wave velocity, are obtained. The Hopf bifurcations in the neighbourhood of the critical points are not found according to the analysis of eigenvalues complexity on these points. Both nonlinear and linear stability analysis show that short and long wave in the liquid film contribute instability and stability respectively. The long wave equation is incompetent to described the free falling film flow under larger Reynolds number and the traveling method is a valid one for simplifying partial differential equations.