1920-21.] Stable Flow of a Fluid in a Uniform Channel. 139 
would immediately be formed at the two edges, of opposite signs, and 
from symmetry necessarily of equal strengths. It will be presumed that 
the whole disturbance is initially concentrated in these two vortices. 
§ 5. Two difficulties immediately present themselves. In the first place, 
if the vorticity thus imposed on the fluid be imagined as concentrated at 
the two points, the condition that there is no slipping at the boundary, as 
will shortly become apparent, is immediately violated. Whether or not this 
is a serious deviation from the real physical conditions is not even yet quite 
definite. Ample experimental evidence exists to show that for moderately 
small speeds — that is to say, for values of U Ijv well below the critical — there 
does not exist any appreciable relative motion at the surface between the 
body and the moving fluid, and this condition is satisfied by the solution 
for steady motion, u — U(1 — y 2 /a 2 ), which has been presumed. For 
velocities in the neighbourhood of and above the critical, however, where 
turbulent motion sets in, experimental results are not so conclusive, and 
do not appear to provide sufficiently definite evidence beyond the fact that 
as the surface is approached the relative velocity does not drop very 
rapidly except when a very close approach is made to the surface. It is, 
in fact, the extremely short distance from the surface within which all 
the fall is to take place that constitutes the real experimental difficulty. 
Whether or not the assumption that a small amount of slipping does take 
place is a serious cause of discrepancy will, I hope, be discussed in a later 
paper, when some of the results of the present discussion will be developed ; 
but in any case it will become evident that the slipping that results from 
the assumption of concentrated vortices is comparatively small in general, 
although in certain circumstances it may be considerable. 
§ 6. The second difficulty is of a different nature, and is not vital. A 
vortex formed in a viscous fluid will not for long maintain its spin 
unimpaired, because of the viscous action of the fluid ; its energy will be 
gradually dissipated into heat. A consideration of the rate of decay of a 
single vortex, say, along the axis of a circular cylinder filled with a viscous 
fluid indicates, in fact, that for a fluid of such small viscosity as water, for 
example, or even air, the rate of decay is very small, and that as far as any 
consideration of the effects immediately subsequent to the formation of the 
vortex is concerned, no serious error will be involved by the assumption 
that k remains constant. This may be illustrated in the following manner. 
The general equations of motion of a viscous fluid in two dimensions 
are known to be 
dt „ K 
( 3 ) 
