Contribution to the theory of cellular thermal convection
Contribution to the theory of cellular thermal convection
复制标题
对细胞热对流理论的贡献
DOI：
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发表时间：
1964
影响因子：
3.7
通讯作者：
Henry Øiann
中科院分区：
文献类型：
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作者：
E. Palm;Henry Øiann
在之前的一篇关于细胞热对流的论文(Palm 1960)中，指出了运动粘度的温度变化所引起的影响的重要性。结果表明，由于非线性相互作用，这种效应将导致六角晶胞的倾向。为了简化数学计算，只考虑了两个波分量之间的相互作用。Segel&Stuart(1962)使用相同的方程，研究了各种均衡解的稳定性。他们得出了一个重要的结论，即对应于六边形的解稳定的必要条件是粘度随温度的变化足够大。在本文件中，从更一般的角度讨论了这一问题。首先，证明了当粘度随温度的变化足够大时，如果只考虑两个波分量，对应于六边形的解是唯一稳定的解。为了检验当运动由任意数目的波分量组成时，这一结果是否也成立，我们研究了三个波分量的情况。事实证明，在这种情况下，唯一可能的模式是由六边形组成的图案。这个结果的有效性很容易推广到更一般的一类波分量。结果表明，对于所有可能发生的小扰动，对应于六边形的解是稳定的。为了证明这一点，有必要考虑非线性扰动理论。从Segel&Stuart的论文和本论文中得到的一个合理的结论是，只有当满足形式(6.9)的条件时，才能观察到六边形图案。然而，关于这个问题的实验还很缺乏。
In a previous paper on cellular thermal convection (Palm 1960) the importance of the effect caused by temperature variation of kinematic viscosity was pointed out. It was demonstrated that this effect would, owing to non-linear interactions, lead to a tendency towards hexagonal cells. For mathematical simplicity, only the interaction of two wave-components was taken into account. Segel & Stuart (1962), working with the same equations, have examined the stability of the various equilibrium solutions. They arrive at the important conclusion that a necessary condition for the solution corresponding to hexagons to be stable is that the variation of viscosity with temperature be sufficiently great. In the present paper the problem is discussed from a somewhat more general point of view. First it is shown that, when the variation of viscosity with temperature is sufficiently great, the solution corresponding to hexagons is the only stable one if only two wave-components are taken into account. To examine if this result is also true when the motion consists of an arbitrary number of wave-components, the case of three wave-components is studied. It turns out that in this case also the only possible mode is the pattern consisting of hexagons. The validity of this result is easily extended to a more general class of wave-components. It is shown that the solution corresponding to hexagons is stable for all small disturbances which can possibly occur. To prove this it is necessary to take into account non-linear disturbance theory. A reasonable conclusion from the paper by Segel & Stuart and the present paper is that a hexagonal pattern is observed only when a condition of the form (6.9) is fulfilled. Experiments concerning this problem are, however, lacking.