Turbulent Flow in Pipes and Channels. 323 



general scheme of hydro-dynamical equations seems to have 

 first been effected by Navier : and Poisson 2 , the justification 

 for this procedure resting on theoretical considerations as to 

 the mutual interactions of the ultimate molecules of fluid. 



Since dXJ/dz represents the rate at which shearing strain 

 is produced by the shearing stress, the hypothesis (1) may 

 be expressed in the language or! elastic solid theory by saying 

 that the ratio of the shearing stress to the rate at which 

 shearing strain is produced is equal to /jl, the coefficient of 

 viscosity. From this point of view the familiar equations 

 of viscous-fluid theory were developed by Saint- Venant 3 

 and Stokes 4 ; from this mode of presentation it appears that 

 (1) is the simplest hypothesis consistent with the linearity 

 of the general equations. 



The experimental justification of the accuracy with which 

 the fundamental hypothesis is able to give an interpretation 

 of reality rests on the comparison of a comparatively small 

 number of simple experimental conditions of viscous flow 

 with the theoretical solutions. Some of these will now be 

 considered in the following sections . 



Section 2. Steady Motion between Parallel Planes. 



The origin is taken half way between the planes and the 

 velocity U is measured along the axis of x parallel to the 

 planes, while the axis of y is taken perpendicular to the 

 planes. If "dpfdtV is the pressure-gradient along the axis 

 of #, the equation of steady motion is 6 



M 3y ,_ B* (2) 



As long as the flow is laminar there is no component of 

 velocity parallel to the y-axis, and 'dpfdx is an absolute 

 constant given by 



'dpfda:=(p s -p 1 )/l, (3) 



where p- 2 — p\ is the pressure-difference between two points 

 at a distance I apart. Assuming that the coefficient /x is an 

 absolute constant depending only on the properties of the 



1 Navier, M&m. de VAcad. des Sciences, t. vi. p. 389 (1822). 



2 Poisson, Journ. de VEcole Polytechn. t. xiii. p. 1 (1829). 



3 Saint-Venant, Comptes Rendus, t. xvii. p. 1240 (1843). 



4 Stokes, Trans. Camb. Phil. Soc. vol. viii. p. 287 (1845) ; Math, and 

 Phys. Papers, vol. i. p. 75. 



5 A valuable historical and critical account of viscosity is given by 

 Brillouin (Marcel), ' Lecons sur la viscosite des liquides et des gaz,' 

 Gauthier-Villars, Paris, 1907. 



6 Lamb, • Hydrodynamics/ p. 542 (1906). 



Z2 



