G. I. TAYLOE ON EDDY MOTION IN THE ATMOSPHERE. 23 



NOTE ON THE STABILITY OF LAMINAR MOTION OF AN INVISCID FLUID, 



MAY 26TH. 



The equation (8) throws a new light on the much discussed question of the stability 

 of the laminar motion of an inviscid fluid. 



Lord RAYLEIGH has considered the stability of a fluid moving in such a way that U, 

 the undisturbed velocity, is parallel to the axis of x and is a function of z. His method 

 is to impose a small disturbing velocity of a type which is simple harmonic with respect 

 to x, satisfies the equations of motion, and contains a factor e wt . He then discusses 

 the conditions under which n may be complex. If n is not complex the motion is 

 stable ; if n is complex the motion is exponentially unstable. 



Perhaps the most important result of Lord RAYLEIGH'S investigation is the conclu- 

 sion he arrives at that if d~~U/dz 2 does not change sign in the space between any two 

 bounding planes, unstable motion is impossible. A particular case of laminar motion 

 in which ePU/cfe 8 has the same sign throughout the fluid is that of an inviscid fluid 

 flowing with the same velocity as a viscous liquid moving under pressure between two 

 parallel planes. In this case, therefore, unstable motion should be impossible. 

 OSBORNE REYNOLDS, however, working in an entirely different way, has come to the 

 conclusion that a viscous fluid moving between parallel planes is unstable if the 

 coefficient of viscosity is less than a certain value which depends on the distance between 

 the planes and on the velocity of the fluid. REYNOLD'S result is in accordance with 

 our experimental knowledge of the behaviour of actual fluids. 



It is evident that there is a fundamental disagreement between the two results for, 

 according to REYNOLDS, the more nearly inviscid the fluid, the more unstable it is 

 likely to be ; while according to RAYLEIGH instability is impossible when the fluid is 

 quite inviscid. 



Various attempts have been made to find the cause of the disagreement, but none 

 of them appear to have been very successful. 



The object of this note is to show that equation (8) may be used to prove the truth 

 of Lord RAYLEIGH'S result for the case of a general disturbance, not necessarily harmonic 

 with respect to x ; and to show also that it may be used to assign a reason for the 

 difference between RAYLEIGH'S and REYNOLDS' results. 



Starting from the principle that when an inviscid fluid in laminar motion is disturbed 

 by dynamical instability, each portion of it retains the vorticity of the layer from which 

 it started, it was shown* that the rate at which momentum parallel to the axis of x 

 flows into a slab of area A and thickness Sz is : 



h^ JJ 



A 



the integrals being taken over the area A. 



* See p. 13. 



