Turbulent Flow in Pipes and Channels. 331 



though some factor contributed to increase very materially 

 the ordinary kinetic-theory coefficient of viscosity. 



As the peculiarity (i.) would seem to associate the pheno- 

 menon with transverse stationary sound-waves (in the 

 particular instance quoted the pitch of the gravest mode 

 would be much above audition), it would seem not unnatural 

 to attribute (ii.) to the same cause. Making use of a 

 conception which appears to have been first put forward by 

 Brillouin 25 in another connexion, the "rationale" of the 

 phenomenon would appear to be as follows : — When flow of 

 a compressible fluid takes place under conditions which 

 permit of the establishment of stationary modes of sound- 

 vibrations (as in flow in non-capillary tubes, channels, &c), 

 the wave-front is refracted by the velocity-gradients which 

 exist in the medium. As rate of change of momentum is 

 propagated along a sound-ray, it follows that, under these 

 conditions, the existence of compressional vibrations will 

 result in a component of rate of change of momentum pro- 

 portional to the velocity-gradient being transferred across 

 an element of surface parallel to the direction of flow. 

 Hence we must write for the viscosity, fM = fx + /u s , where /jl 

 is the ordinary viscosity of the kinetic theory, and /a, is a 

 " quasi-vi^cosity " due to the existence of compressional 

 waves and is proportional to the intensity of the wave-motion. 

 The theoretical velocity-gradients calculated on this theory 

 give a very satisfactory account of the experimental results : 

 the full details of the theory and the comparison with 

 experimental results will be given in Part III. of the present 

 paper. 



It thus appears that the gradual modification with 

 increasing velocities of the laminar flow in channels throuoh 

 the interaction of transverse sound-vibrations with viscous 

 tractions is an essential factor in ultimately leading to the 

 breakdown of stream-line motion. It appears to the writer 

 that in the combination of compressional vibrations and 

 shearing-motion lies the origin of a vortex rotation which, 

 as the amplitudes of the sound-vibrations and their frequencies 

 increase, may ultimately result in the formation of finite 

 vortex filaments, resulting finally in "turbulent motion" in 

 the generally accepted sense of the word. 



As the factor of compressibility, hitherto neglected in 

 theoretical treatments of the stability of laminar flow by 

 Reynolds, Rayleigh, Kelvin, Sharpe, Orr, and others, 

 appears to be of considerable importance in the light of the 



25 Brillouin, L., " Conduotibilite calorifique et viscosite des liquides 

 monoatoiniques," Comptes Rendus, vol. clix. pp. 27-30, July 6th, 1914. 



