437° The Stability of Fluid Motion. 
wall, together with no tangential slipping; in fact, his boundary conditions 
come under case (i1) of § 4, namely u = 0, p,, = 0. With these assumptions, 
he applies the method of small vibrations to simple shearing motion between 
a fixed plane and a parallel moving plane. It appears that the motion is 
unstable for disturbances whose wave-length exceeds a certain value; for 
smaller wave-lengths it is stable or unstable according to the value of R. 
Thus the motion is not thoroughly stable. Without discussing how far these 
assumptions express the behaviour of actual fluids and boundaries, we may 
see how they affect the enerey method. 
We shall take the case of laminar flow between fixed planes, for which the 
previous calculations are available. 
The stream function y satisfies equation (11), and the boundary conditions 
are 
== 0s —p+2pdv/dy = 0. 
From the equations (1), these are equivalent to 
= 03 pv dU /dy—pPu/oy? = 0, 
or, in the present notation, 
dyy/da = 0; dp [da — harp =0): a ip. (26) 
Using the solutions yy and yo, these give 
wo (ra — 2kprr2)— pa! (ro — 2kp ro) = 0, (27) 
where accents denote differentiation with respect to «. 
From the previous work, this equation involves odd powers of k. But & is 
37R/4p? and we have to determine R in terms of p from (27). It follows that 
in this case there is no real solution of the problem of finding the critical 
minimum of R. 
It seems probable that it is only those motions which are completely stable 
in the ordinary theory which lead also to a real minimum for R. The suggestion 
may be stated in this manner: if a fluid motion is thoroughly stable when 
considered by the method of small vibrations applied to equation (2) and the 
boundary conditions, then it also possesses a real minimum value of R found 
from equation (7) and the boundary conditions. It has been pointed out 
that the latter equation is derived directly from the former, and it may be 
presumed that the minimum value of R depends in some manner upon the 
rates of decay of elementary vibrations and so may be used as a measure of 
the degree of stability of the system. 
Harrison anp Sons, Ltd., Printers in Ordinary to His Majesty, St. Martin’s Lane, 
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