On the Motion of Paired Vortices with a Common Axis
On the Motion of Paired Vortices with a Common Axis
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关于共轴成对涡旋的运动
DOI：
10.1112/plms/s1-25.1.185
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发表时间：
1893
影响因子：
1.8
通讯作者：
A. Love
中科院分区：
文献类型：
--
作者：
A. Love
1.本文的研究是为了进一步阐明涡环运动中的一个问题，这个问题是Helmholtz在他最初的涡旋运动回忆录中首次考虑的。*他发现，具有相同轴线和相同方向的两个涡环沿着平行于该轴的同一方向运动；“最前面的涡环变宽和移动得更慢，追赶者收缩和移动得更快，直到最后，如果它们的速度相差不太大，它就会超过第一个涡环并穿透它。然后，同样的游戏以相反的顺序进行，因此两个涡环交替通过。”F对这里所述的动议进行更详细的说明是极其困难的。“我们不知道运动可能是周期性的条件，当未知条件满足时，我们只能猜测周期的长度。然而，在将涡旋-原子理论应用于辐射和化学结合问题时，可以想象，这一周期和运动类型可能会起到重要作用。在这里，我建议通过考虑无限大流体中存在两对无限小截面的柱状涡的情况来模拟该问题的一些情况，其中每一对的两个涡的环流是相等的且具有相反的符号，围绕四个涡的环流在绝对大小上是相等的，并且其中一对的对称线与另一对的对称线重合。这种类型的单个对以恒定速度平行于对称轴运动。在适当的方向上循环的两对相互影响对方的运动，其方式类似于细环的运动。我发现了一个条件，即运动可能是周期性的，周期的长度，以及由一对涡流相对于另一对涡流的相应涡流所描述的曲线的形状。
1. The investigation in this paper was undertaken with the view of throwing some further light on a problem in the motion of vortexrings which was first considered by Helmholtz in his original memoir on vortex-motion.* He found that two vortex-rings having the same axis and circulations in the same direction travel in the same direction parallel to the axis;" the foremost widens and travels more slowly, the pursuer shrinks and travels faster, till finally, if their velocities are not too different, it overtakes the first and penetrates it. Then the same game goes on in the opposite order, so that the rings pass through each other alternately." f It is extremely difficult to obtain a more detailed account of the motion here described." We are ignorant of the condition that the motion may be periodic, and we can only make guesses at the length of the period when the unknown condition is satisfied. Yet in applications of the vortex-atom theory to problems of radiation and chemical combination, it is conceivable that this period and the type of motion may play an important part. I propose here to imitate some of the circumstances of the problem by considering the case where there are present in an infinite fluid two pairs of cylindrical vortices of indefinitely small section, the circulations about the two vortices of each pair being equal and of opposite sign, the circulations about the four vortices being equal in absolute magnitude, and the line of symmetry for one pair coinciding with that for the other. A single pair of this kind moves parallel to the axis of symmetry with constant velocity. Two pairs with circulations in the proper directions influence each other's motions in a manner analogous to that exhibited by thin rings. I find a condition that the motion may be periodic, the length of the period, and the form of the curve described by one vortex of one pair relative to the homologous vortex of the other pair.