268
ON THE STABILITY OF VISCOUS FLUID MOTION
[388
re-established by slowing down and bringing up to speed again, gradually enough.
" Without the disturbing rod at all, I found that by resisting the outer can by hand somewhat suddenly, but not very much so, the torque increased suddenly and the motion became visibly turbulent at the lower speed and remained so.
" I have no doubt we should find with higher and higher speeds, very gradually reached, stability of laminar or non-turbulent motion, but with narrower and narrower limits as to magnitude of disturbance; and so find through a large range of velocity, a confirmation of Phil. Mag. 1887, 2, pp. 191—196. The experiment would, at high velocities, fail to prove the stability which the mathematical investigation proves for every velocity however high.
water-
mercury-
hting
tofsionally
to measure
torque
-L
rotating
rotating fixed
Fig. 1.
"As to Phil. Mag. 1887, 2, pp. 272—278,1 admit that the mathematical proof is not complete, and withdraw [temporarily ?] the words ' virtually inclusive' (p. 273, line 3). I still think it probable that the laminar motion is stable for this case also. In your (Phil. Mag. July 1892, pp. 67, 68) refusal to admit that stability is proved you don't distinguish the case in which my proof was complete from the case in which it seems, and therefore is, not
complete.
v
" Your equation (24). of p. 68 is only valid for infinitely small motion, in which the squares of the total velocities are everywhere negligible; and in this case the motion is manifestly periodic, for any stated periodic conditions of the boundary, and comes to rest according to the logarithmic law if the boundary is brought to rest at any time.of stability by the second method, and Orr§ has since shown that the same objection applies to the special method. Accordingly Kelvin's proof of stability cannot be considered sufficient, even in case (i). That Kelvin himself (partially) recognized this is shown by the following interesting and characteristic letter, which I venture to give in full.