The Stability of Fluid Motion. 430 
Reynolds’ method of using (5) to determine a criterion of stability 
eonsisted, in assuming a suitable form for x and finding the least value of It 
for which the right-hand side of (5) is zero. It is usually stated that this 
method assumes turbulent motion to be already in existence, and it then 
gives a criterion to show whether the turbulence is increasing or decreasing 
momentarily ; but this is somewhat misleading without defining what is 
meant by turbulent motion. Equation (5), as stated above, applies to any 
small arbitrary disturbance, neglecting terms of the second order, as in the 
ordinary method of small vibrations; further, U is a laminar fluid motion 
satisfying the usual hydrodynamical equations under the given conditions. 
On the other hand, Reynolds defined U as the mean velocity at each 
point, taken over a small region or during a short time, and this principal or 
mean motion need not satisfy the ordinary equations. The extra velocities 
uw and v then play a double part, in that they specify the disturbance, and at 
the same time give a measure of the turbulence; they must satisfy certain 
conditions as to their mean values, and then equation (5) holds in the same 
form when mean values are used. However, in applying it to find the 
criterion for flow under a constant pressure gradient, Reynolds, and Sharpe 
following him, did, in fact, take U to be the usual form, C (a?—y?), for steady 
laminar flow. But in turbulent flow, although the variations of velocity at 
any point are small, yet they may cause the gradient of the mean velocity to 
differ appreciably from its value in laminar flow, as is obvious from a com- 
parison of the curves of distribution of velocity across a pipe in regular and 
in turbulent flow. 
” However, it is unnecessary to dwell on this distinction, as it has been 
pointed out clearly by Lorentz* and other writers; further, we shall 
consider here only small disturbances. 
3. Under these circumstances, the energy method has been given a precise 
and definite meaning by Orr} from the following considerations :— 
If the right-hand side of (5) is positive, the energy of the disturbance is 
momentarily increasing. But, for a given velocity distribution, U, it may be 
impossible to find any function, y, satisfying the boundary conditions, such 
that that expression is positive, unless R exceeds a certain value. If such be 
the case, this least value of R is a critical value of definite significance. The 
corresponding critical disturbance is found by taking the variation of the 
Soman R | | U x dé dn —2 {| (yp dé dn = 0, (6) 
subject to dR = 0. 
* H. A. Lorentz, ‘Abhandlungen iiber Theor. Phys., vol. 1, p. 43. 
+ W. MeF. Orr, ‘ Proc. Roy. Irish Acad.,’ vol. 27, p. 9 (1907). 
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