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XLII. Note on the Flow of Water in a Straight Pipe. 

 By M. P. Eudski, Priv. Doc. in the University of Odessa *» 



IT is a known fact that the law of resistance to the motion 

 of a liquid in pipes and channels of great size differs much 

 from that in capillary tubes. It is also known that this differ- 

 ence is due to the presence of eddies in great pipes, while in 

 capillary tubes the liquid flows in straight lines. Prof. Osborne 

 Keynolds t has shown that there exists a certain critical mean 

 velocity, depending on the diameter of the pipe and on 

 the temperature (i. e. viscosity) at which the eddies must 

 appear. He thinks that the appearance of eddies is due to 

 the instability of rectilineal motion. But Lord Kelvin J has 

 shown that at least for small disturbances the rectilineal 

 motion is stable provided the coefficient of viscosity is not zero. 

 Although Lord Rayleigh § thinks that Lord Kelvin's proof is 

 not quite convincing, it seems to me to be so, because the 

 steady rectilineal motion with zero velocity at the walls 

 satisfies the condition that the loss of energy shall be the 

 least possible. This motion belongs to the type which was 

 shown by Helmholtz to have this property || . Now it is known 

 that generally the motions, which in a certain manner satisfy 

 the minimum or maximum condition, are stable. 



The same question was also treated by Mr. Basset H". He 

 has found the rectilineal motion unstable. As far as I can 

 understand him, from a short communication, it was only 

 after he had introduced in the expression of resistance a term 

 depending on the square of relative velocity. In doing so he 

 has anticipated the law of resistance proper to the eddying 

 motion. On the other hand, he finds that without this term 

 the steady rectilineal motion remains always stable. His 

 results also agree closely with the results of Lord Kelvin. 



It seems to me that all this clearly agrees in showing 

 that it is not the question of stability or instability which 

 arises here, but another one. In speaking of stability we 

 mean eo ipso the tacit assumption that the eddying motion 

 may be also expressed with the help of functions satisfying 

 the common partial differential equations of viscous fluid 



* Communicated by the Author. 

 f Phil. Trans. 1883, p. 935. 

 X Phil. Mag. 5 ser. xxiv. pp. 188 and 272. 

 § Phil. Mag. 5 ser. xxxiv. p. 67, 

 || Basset, Hydrodynamics, vol. ii. p. 356. 

 il Proceedings Roy. Soc. vol. hi. no. 317, p. 27-*>. 



