基于运算微积的Stokes第二问题起动过程的精确解

为研究Stokes第二问题的起动过程,利用运算微积得到了问题的精确解。在时间趋于无限长时,该解逼近Stokes第二问题的精确解。研究发现:空间每一点的速度从初始时刻的零值开始增大,达到第一峰值后开始减小并进入振荡状态;流动开始为非等幅振荡,随时间进程最终发展为稳定的等幅振荡。在近壁面处,第一峰值低于该处达到稳定等幅振荡后的幅值;在远壁面处,第一峰值高于该处的等幅振荡幅值;之间存在一个临界距离,该处的第一峰值与等幅振荡的幅值相等。此外,平板振荡频率越低,流场达到稳态振荡所需时间越长,临界距离越短。研究还给出了频率和临界距离的定量关系,发现两者的乘积为一常数。

Exact solution of Stokes second problem including startup process by using operational calculus

The exact solution of the Stokes second problem including the startup process was obtained using operational calculus. For very long time, the present exact solution agrees with the classical solution of steady-state Stokes second problem. The exact solution showed that the velocity of a point in space increases from zero and the flow from unequal oscillation amplitudes transitions to equal oscillation amplitudes with increasing time. Near the plate, the first peak value is always lower than the one of the equal amplitude oscillations at the same place. As the distance from the plate increases, the peak value becomes higher than the one of the equal amplitude oscillations. It is interesting that there is a critical distance from the plate, where the first peak is equal to the one of the equal amplitude oscillations. The critical distance is shorter at higher frequencies. The numerical results show that the product of the frequency and the critical distance is a constant.