标题： 莫法特-普克纳切夫流：一个旧问题的新转折 显示英文标题

标题： The Moffatt-Pukhnachev flow: a new twist on an old problem

摘要： 在水平圆柱体外表面，其角速度为具有指定频率和振幅的周期性时间变化的情况下，研究了薄粘性薄膜的流动。 恒定角速度问题最初由Moffatt（1977）和Pukhnachev（1977）研究。 忽略表面张力。 在一定范围的振荡振幅和频率下，求解了膜厚度的演化方程。 一个在振幅-频率空间中绘制的爆裂图显示了高度复杂的类似分形的结构，表现出自相似性。 对于一般的初始条件，薄膜表面在有限时间内达到斜率奇点并趋于翻转。 使用多尺度方法对高频和低频极限进行了渐近分析。 在高频情况下，分析表明，适当选择初始轮廓可以显著延迟翻转时间，甚至产生周期性时间解。 在低频极限下，如果振荡振幅低于临界值，可以构造一个不翻转的准周期解。 超过该值后，解不可避免地趋向于爆裂。 展示了如何与稳定旋转圆柱体问题共同构造出现单个冲击或双冲击的解。 显示英文摘要

摘要： The flow of a thin viscous film on the outside of a horizontal circular cylinder, whose angular velocity is time-periodic with specified frequency and amplitude, is investigated. The constant angular velocity problem was originally studied by Moffatt (1977) and Pukhnachev (1977). Surface tension is neglected. The evolution equation for the film thickness is solved numerically for a range of oscillation amplitudes and frequency. A blow-up map charted in amplitude-frequency space reveals highly intricate fractal-like structures exhibiting self-similarity. For a general initial condition the film surface reaches a slope singularity at a finite time and tends to overturn. The high-frequency and low-frequency limits are examined asymptotically using a multiple-scales approach. At high frequency the analysis suggests that an appropriate choice of initial profile can substantially delay the overturning time, and even yield a time-periodic solution. In the low-frequency limit it is possible to construct a quasi-periodic solution that does not overturn if the oscillation amplitude lies below a threshold value. Above this value the solution tends inexorably toward blow-up. It is shown how solutions exhibiting either a single-shock or a double-shock may be constructed in common with the steadily rotating cylinder problem.