1913] ON THE MOTION OF A VISCOUS FLUID 193
If the superposed motion also be two-dimensional, it may be expressed by means of a stream-function ty. We have in terms of polar coordinates
, d^r d-^r I d^lr
u = -j1- = -T- sm. 0 H --- =-£• cos 6, dy dr r d&
, dty d-*lr , I d^!r . n ~v = -T- = -T- cos 6 --- JTT sin 0, dx dr r dd
so that
u'2 - 1 ;'2 = (cos2 0 — sin2 0) •< [ ) 1 ) r -l- i sin 0 cos 0 cZi/r d^
( \ U/ 1/ / \ U// / J r d»- d9 '
_ , vf = cos 0 sin i f/dtyy 1 /ctyy| cos20 — sin2 0 (Z'xjr cZi|r
(\dr) r*\d6)} + r dr dd
Thus
cos 6 sin 0 ( u"* - v*2) - (cos2 0 - sin2 0) wV I d^d^ (26)
r dr d6'
and (25) becomes
dT'
T', F', as well as the last integral, being proportional to z.
We suppose the motion to take place in the space between two coaxal cylinders which revolve with appropriate velocities. If the additional motion be also symmetrical about the axis, the stream-lines are circles, and ty is a function of r only. The integral in (27) then disappears and dT'jdt reduces to — F', so that under this restriction * the original motion is stable. The experiments of Couettef and of MallockJ, made with revolving cylinders, appear to show that when w', v', w' are not specially restricted the motion is unstable. It may be of interest to follow a little further the indications of (27).
The general value of ^ is
T^= C0 + Ci cos 0 + $ sin 0 + ... + Cn cos n0 + Snsmnd, ......(28)
Gn, Sn being functions of r, whence
n being I, 2, 3, &c. If Sn, Gn differ only by a constant multiplier, (29) vanishes. This corresponds to
•^r = .Bo + RI cos (6 + ej) + ... + Rn cos n (d + en) + ..., ......(-30)
* We may imagine a number of thin, coaxal, freely rotating cylinders to be interposed between the extreme ones whose motion is prescribed, t Ann. d. Chimie, t. xxi. p. 433 (1890). t P?-oc. Roij. Soc. Vol. MX. p. 38 (1895).
R. VI. 13both motions satisfying the dynamical equations, and giving the prescribed boundary velocities; and we consider the expression for the kinetic energy T' of the motion (1) which is the difference of these two, and so makes the velocities vanish at the boundary. The motion M' with velocities u', v', w' does not in general satisfy the dynamical equations. We have