1883.J 



On the Motion of Water. 



95 



13. Further Study of the Equations of Motion. — Having now definite 

 data to guide me, I was anxious to obtain a fuller explanation of these 

 results from the equation of motion. I still saw only one way open 

 to account for the instability, namely, by assuming the instability of a 

 frictionless fluid to be general. 



Having found a method of integrating the equations as far as to 

 show whether any particular form of steady motion is stable for a 

 small disturbance, 1 applied this method to the case of parallel flow 

 in a frictionless fluid. The results which I obtained at once were, 

 that flow in one direction was stable, flow in opposite directions 

 unstable. This was not what I was looking for, and T spent much 

 time in trying to find a way out of it, but whatever objections my 

 method of integration may be open to, I could make nothing less 

 of it. 



It was not until the end of 1 882 that I abandomed further attempts 

 with a frictionless fluid and attempted by the same method the integra- 

 tion of a viscous fluid. This change was in consequence of a dis- 

 covery that in previously considering the effect of viscosity I had 

 omitted to take fully into account the boundary conditions which 

 resulted from the friction between the fluid and the solid boundary. 



On taking these boundary conditions into account, it appeared that 

 although the tendency of viscosity through the fluid is to render 

 direct or steady motion stable, yet owing to the boundary condition 

 resulting from the friction at the solid surface, the motion of the fluid 

 irrespective of viscosity would be unstable. Of course this cannot 

 be rendered intelligible without going into mathematics. But what I 

 want to point out is that this instability, as shown by the integration 

 of the equations of motion, depends on exactly the same relation 



P c 



as that previously found. 



This explained all the practical anomalies, and particularly the 

 absence of eddies below a pure surface of water exposed to the wind ; 

 for in this case, the surface being free, the boundary condition was 

 absent, whereas the film of oil by its tangential stiffness introduced 

 this condition. This circumstance alone seemed a sufficient verifica- 

 tion of the theoretical conclusion. 



But there was also the sudden way in which eddies came into 

 existence in the experiments with the colour band, and the effect of 

 disturbances to lower the critical velocity. These were also explained, 

 for as long as the motion was steady the instability depended upon the 

 boundary action alone, but once eddies introduced the stability would 

 be broken down. 



It thus appeared that the meaning of the experimental results had 



