WAVE FORMATION IN LAMINAR FLOW DOWN AN INCLINED PLANE
WAVE FORMATION IN LAMINAR FLOW DOWN AN INCLINED PLANE
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DOI：
10.1017/s0022112057000373
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发表时间：
1957-01-01
影响因子：
3.7
通讯作者：
BENJAMIN, TB
中科院分区：
文献类型：
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作者：
BENJAMIN, TB
本文从理论上讨论了雷诺数R较小的流体动力稳定性问题。其稳定性进行检查的主流由一个均匀的层流的粘性液体运行下的倾斜平面的作用下的重力，被限制在一侧的自由表面的表面张力的影响。因此，这个问题直接关系到在化学工程中有重要用途的薄液膜的性质。过去的许多实验表明，在沿壁向下流动时，除了R很小的时候，流体明显地被波搅动;例如，在垂直的水膜上，可以观察到波，直到R减小到某个值而不是小于10。因此，目前的治疗方法是基于近似适合于相当低的R值，从而避免了严重的数学困难通常在高R稳定性问题。这个问题的提法类似于Yih（1954）的提法;但解决的方法与他的不同，各自的结果是矛盾的。特别是在R很小的情况下，严格垂直流的稳定性问题上存在着分歧。与以前的结论相反，这里表明流动总是不稳定的：即对于R的所有有限值，存在一类无阻尼波。然而，当R相当小时，不稳定波的放大率变得非常小，它们的波长变得非常大;这为某些实验观测中明显没有波提供了一个令人满意的解释，也为现有的R的“准临界”值估计中的广泛分散提供了一个令人满意的解释。鉴于这些结果的争议性质，重点是给目前的工作和已建立的滚波理论之间的各种协议点;后一种理论给出了引力流上波形成的物理机制的清晰图像，根据这一点，本文的结果是完全合理的。我们把控制中性稳定性的条件推到了三阶，参数为显示为小;但是一个不太精确的近似值被证明是一个容易操作的理论的充分基础，提供了一个现成的实验检验。这个理论被用来预测R值，在该值下，可观察到的波浪应该首先在垂直水膜上发展，以及波浪的长度和速度。这三个预言与Binnie（1957）的实验结果进行了比较，并得到了实质性的证实。
This paper deals theoretically with a problem of hydrodynamic stability characterized by small values of the Reynolds number R. The primary flow whose stability is examined consists of a uniform laminar stream of viscous liquid running down an inclined plane under the action of gravity, being bounded on one side by a free surface influenced by surface tension. The problem thus has a direct bearing on the properties of thin liquid films such as have important uses in chemical engineering.Numerous experiments in the past have shown that in flow down a wall the stream is noticeably agitated by waves except when R is quite small; on a vertical water film, for instance, waves may be observed until R is reduced to some value rather less than 10. The present treatment is accordingly based on methods of approximation suited to fairly low values of R, and thereby avoids the severe mathematical difficulties usual in stability problems at high R. The formulation of the problem resembles that given by Yih (1954); but the method of solution differs from his, and the respective results are in conflict. In particular, there is dis-agreement over the matter of the stability of a strictly vertical stream at very small R. In contrast with the previous conclusions, it is shown here that the flow is always unstable: that is, a class of undamped waves exists for all finite values of R. However, the rates of amplification of unstable waves are shown to become very small when R is made fairly small, and their wavelengths to become very large; this provides a satisfactory explanation for the apparent absence of waves in some experimental observations, and also for the wide scatter among existing estimates of the ‘quasi-critical’ value of R below which waves are undetectable. In view of the controversial nature of these results, emphasis is given to various points of agreement between the present work and the established theory of roll waves; the latter theory gives a clear picture of the physical mechanism of wave formation on gravitational flows, and in its light the results obtained here appear entirely reasonable.The conditions governing neutral stability are worked out to the third order in a parameter which is shown to be small; but a less accurate approximation is then justified as an adequate basis for an easily workable theory providing a ready check with experiment, This theory is used to predict the value of R at which observable waves should first develop on a vertical water film, and also the length and velocity of the waves. These three predictions are compared with the experimental results found by Binnie (1957), and are substantially confirmed.