中文摘要：

在竖直振动的圆柱形容器中，将Navier-Stokes方程线性化，利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分，对两部分区域分别求解，得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析，得到形成稳定表面波的条件，给出了临界曲线．此外，还获得了阻尼系数的解析表达式．最后，将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正，理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现，当驱动的频率较低时，流体的粘性对表面波模式选择有重要影响，而表面张力的影响不明显；但当驱动频率较高时，流体的表面张力起主要作用，而流体的粘性影响甚小．

英文摘要：

In a vertically circular cylindrical container, singular perturbation theory of twotime scale expansions was developed in weakly viscous fluids to investigate the motion of single free surface standing wave by linearizing the Navier-Stokes equation. The fluid field was divided into an outer potential flow region and an inner boundary layer region. The solutions of both two regions were obtained and a linear amplitude equation incorporating damping term and external excitation was derived, The condition to appear stable surface wave was obtained and the critical curve was determined. In addition , an analytical expression of damping coefficient was determined. Flnally, the dispersion relation, which has been derived from the inviscid fluid approximation, is modified by adding linear damping. It is found that the modified results are more reasonably close to experimental results. Result shows that when forcing frequency is low, the viscosity of the fluid is prominent for the mode selection. However, when forcing frequency is high, the surface tension of the fluid is prominent.