On the Motion of a Viscous Fluid
On the Motion of a Viscous Fluid
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DOI：
10.1080/14786441308635022
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发表时间：
1913-10-01
影响因子：
1.6
通讯作者：
Rayleigh, Lord
中科院分区：
文献类型：
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作者：
Rayleigh, Lord
F= f0 + F' ......(6)或者因为F必然是正的，所以运动Me使F成为绝对最小值。应该指出的是，F'只有在固体（Stokes）所假定的运动中才会消失，而这种运动不会使边界速度消失。由(4)决定的运动M0是唯一的。(6)中关于Me使F为绝对最小的结论，并不局限于慢动作的假设。保证(6)所依赖的(5)的满足所需要的是，V: o,~ 2v0， V: o应该是某个单值函数的导数。显然，~ u0， V2~就足够了。0， V~ 0消失，如果运动有速度势就会发生这种情况。斯托克斯很久以前就说过，当存在速度势时，不仅流体运动的一般方程得到满足，而且考虑摩擦时得到的方程也同样得到满足。总能找到一个具有速度势的运动，它在边界处有规定的法向速度，切向速度就由此确定了。如果这些与规定的粘性流体的切向速度一致，则所讨论的运动满足所有条件。由于这一运动使F成为绝对最小值，所以在相同的边界条件下，它不能与由(4)决定的运动d~不同。考虑一般的运动方程，我们可以得出同样的结论
F= F0+ F',......(6) or since F t is necessarily positive, the motion Me makes F an absolute minimum. It should be remarked that F'can vanish only for a motion such as can be assumed by a solid body (Stokes), and that such a motion could not make the boundary velocities vanish. The motion M0 determined by (4) is thus unique.The conclusion expressed in (6) that Me makes F an absolut6 minimmn is not limited to the supposition of a slow motion. All that is required to ensure the fulfilment of (5), on which (6) depends, is that V: Uo,~ 2v0, V: wo should be the derivatives of some single-valued function. Obviously it would suffice that~ u0, V2~. 0, V~ w0 vanish, as will happen if the motion have a velocity-potential. Stokes* remarked long ago that when there is a velocity-potential, not only are the ordinary equations of fluid motion satisfied, but the equations obtained when friction is taken into account are satisfied likewise. A motion with a velocity-potential can always be found which shall have prescribed normal velocities at the boundary, and the tangential velocities are thereby determined. If these agree with the prescribed tangential velocities of~ viscous fluid, all the conditions are satisfied by the motion in question. And since this motion makes F an absolute minimum, it cannot differ from the motion d~ termined by (4) with the same boundary conditions. We may arrive at the same conclusion by considering the general equation of motion--