[Reprinted from the PROCEEDINGS OF THE Roya Society, A. Vol. 98] 
The Stability of Fluid Motion. 
By T. H. Havetock, F.R.S. 
(Received January 31, 1921.) 
1. The following notes on the stability of fluid motion arose from a desire 
to use the energy method, introduced by Reynolds and moditied by Orr, as a 
measure of the comparative degree of stability of various types of flow under 
different boundary conditions. A few examples are worked out to illustrate 
this point of view: in § 5 a case which resembles the flow of a stream with a 
free surface; in §7 flow which approximates to a uniform stream between 
fixed walls without slipping at the walls; in §§6,8 motion with other 
boundary conditions. Before proceeding to these, it seems desirable to give a 
short account of the method in the form in which it is used later, together 
with some remarks on its relation to the classical method of small vibrations. 
2. We shall consider only two-dimensional motion of an incompressible 
viscous fluid limited by the planes y= +a. Let the steady state under an 
assigned forcive and given boundary conditions be specified by a velocity, U, 
parallel to the axis of wz. Let the disturbed state have velocity components 
(U+4u, v)and let the additional pressure be p. Then, by taking the difference 
of the two sets of hydrodynamical equations for the two states and neglecting 
squares and products of the additional velocities, we have 
ow ou, OU 1 op 
tt UY == Ss BK W2 
eP wag’ foyie pant” 
Ov 77 Ov 1 Op 
AE) Of ee Eee Pp 1 
Bo Bs p yp ar ©) 
together with the equation of continuity. 
It is convenient to introduce non-dimensional variables given by 
Ge = We y = an; at = Ut; 
where U is the mean velocity over the cross-section in the steady state. 
Further, we write UU instead of U, and take the current function of the 
additional velocity to be Uay. Eliminating p from the two equations (1), we 
obtain 
@ , @\ 5 WOOP a 
R (sta) v—RU ae ay, (2) 
where U” is written for d?U/dn?, and R is Reynolds’ number 2aU/v. There 
are in addition the appropriate boundary conditions for the disturbing 
function, y. The classical method of examining the stability of a given 
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