Turbulent Flow in Pipes and Channels. 329 



become narrower as /a diminishes/' These views have been 

 criticized by both Rayleigh :9 and Orr 20 . 



Iiayleigh's earliest papers on the subject deal with the 

 laminar steady motion of a perfect fluid between parallel 

 planes in which the distribution of velocity is continuous, 

 while the rotation or vorticity changes suddenly on passing 

 from one layer of finite thickness to the next. Rayleigh's 

 general conclusion is thus stated : " The steady motion of a 

 non-viscous fluid in two dimensions between fixed parallel 

 walls is stable provided that the velocity U everywhere 

 parallel to the walls is such that d' 2 TJ/di/ 2 is of one size 

 throughout, y being the ordinate measured perpendicularly 

 to the walls. It is here assumed that the disturbance is 

 infinitesimal/' 



The extension of this ideal problem to include the case of 

 an actual fluid possessing viscosity brings us at once to the 

 main (theoretical) difficulty of the subject, and one which 

 cannot yet be regarded as (theoretically) settled. The 

 subject was proposed for discussion in the Adams Prize 

 Essay of 1887, and may be considered to constitute the 

 simplest case of fluid resistance. Many v r riters, including 

 Reynolds himself 21 , have since that date contributed to the 

 theoretical aspect of the subject, and have succeeded by 

 various methods in establishing theoretically determined 

 values of Reynolds's Constant K, for cases of flow which 

 not only include that between parallel planes but also the 

 more difficult one of the circular tube. These are reviewed 

 in an important memoir by Orr 20 , who himself gives an 

 original treatment of the subject. An authoritative account 

 of the present state of the problem, from which the writer 

 has quoted the above passages, has recently been published 

 by Rayleigh 22 . It is therefore unnecessary to deal at 

 greater length with this aspect of the subject in the present 

 paper, beyond stating that the most recent theoretical treat- 

 ment of the subject by Hopf 23 , following a method due to 



10 Rayleigh-, Phil. Mag. vol. xxiv. pp. 59-70 (1892) : Collected Works, 

 vol. iii. p. 575. 



20 Orr, Proc. Roy. Irish Acad. vol. xxvii. pp. 9-138 (1907). This 

 important memoir gives a critical account of the theoretical work on 

 the subject to the year 1907 in great detail. 



21 Reynolds, Phil. Trans, vol. clxxxvi. A, p. 123 (1894): Scientific 

 Papers, ii. p. 535. 



22 Ravleigh, "On the Stability of Viscous Fluid Motion/' Phil. Mag. 

 Oct. 1914 ; see also Phil. Mag. Sept. 1915. 



23 Hopf, Ann. d. Phys. xliv. no. 9, pp. 1-60 (]914). See also a recent 

 discussion of the subject bv Taylor, R. I., Phil. Trans. Rov. Soc 

 vol. ccxv. A. pp. 23-26 (1915). 



