194 ON THE MOTION OF A VISCOUS FLUID [376
where R0> R1} &c. are functions of r, while e,, e2, &c. are constants. If i/r can be thus limited, dT' /dt reduces to -F', and the original motion is stable.
T . dT' -,, 0 [* fa dOn n dSn\dr
In general -r- = F ^irpz lz,n(bn -5 -- Un -r- .......... (31)
° dt ^ J \ dr dr / r2 ^ '
Gn, Sn must be such as to give at the boundaries
Cn=0, dCn/dr = 0, Sw = 0, dSnldr = Q; ............ (32)
otherwise' they are arbitrary functions of r. It may be noticed that the sign of any term in (29) may be altered at pleasure by interchange of Gn and Sn.
When /i is great, so that the influence of F preponderates, the motion is stable. On the other hand when //, is small, the motion is probably unstable, unless special restrictions can be imposed.
A similar treatment applies to the problem of the uniform shearing motion of a fluid between two parallel plane walls, defined by
Wo = 0, w0 = 0 ...................... (33)
From (23) =- = -F'-pB u'v'dxdy ...................... (34)
Cut J J
If in the superposed motion vf = 0, the double integral vanishes and the original motion is stable. More generally, if the stream-function of the superposed motion be
^ = C cos kx + S sin kx, ........................ (35)
where C, S are functions of y, we find
......... . ...... (36)
dy) '' x '
Here again if the motion can be such that 0 and S differ only by a constant multiplier, the integral would vanish. When ^ is small and there is no special limitation upon the disturbance, instability probably prevails. The question whether. /* is to be considered great or small depends of course upon the other data of the problem. If D be the distance between the planes, we have to deal with BD^-jv (Reynolds).
In an important paper * Orr, starting from equation (34), has shown that if BDz(v is less than 177 "every disturbance must automatically decrease, and that (for a higher value than 177) it is possible to prescribe a disturbance which will increase for a time." We must not infer that when
* Proc. Roy. Irish Acad. 1907. . . ' he 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