1894.] Theory of Incompressible Viscous Fluids, fyc. 41 



hand, these results, both theoretical and practical, were directly at 

 variance with common experience as to the resistance encountered by 

 larger bodies moving with higher velocities through water, or by 

 water moving with greater velocities through larger tubes. This dis- 

 crepancy Sir G. Stokes considered as probably resulting from eddies 

 which rendered the actual motion other than that to which the sin- 

 gular solution referred and not as disproving the assumption. 



In 1850, after Joule's discovery of the Mechanical Equivalent of 

 Heat, Stokes showed, by transforming the equations of motion with 

 arbitrary stresses so as to obtain the equation of (" Vis-viva ") 

 energy, that this equation contained a definite function, which repre- 

 sented the difference between the work done on the fluid by the 

 stresses and the rate of increase of the energy per unit of volume, 

 which function, he concluded, must, according to Joule, represent the 

 Vis- viva converted into heat. 



This conclusion was obtained from the equations irrespective of any 

 particular relation between the stresses and the rates of distortion. 

 Sir G. Stokes, however, translated the function into an expression in 

 terms of the rates of distortion, which expression has since been 

 named by Lord Bayleigh the Dissipation Function. 



In 1883 the author succeeded in proving, by means of experiments 

 with colour bands the results of which were communicated to the 

 Society* that when water is caused by pressure to flow through a 

 uniform smooth pipe, the motion of the water is direct, i.e., parallel to 

 the sides of the pipe, or sinuous, i.e., crossing and recrossing the pipe, 

 according as U m , the mean velocity of the water, as measured by 

 dividing Q, the discharge by A, the area of the section of the pipe, is 

 below or above a certain value given by KfijDp, where D is the 

 diameter of the pipe, p the density of the water, and K a numerical 

 constant, the value of which according to the author's experiments 

 and, as he was able to show, to all the experiments by Poiseuille and 

 Darcy, is for pipes of circular section between 



1,900 and 2,000, 



or, in other words, steady direct motion in round tubes is stable or 

 unstable according as 



or >2000 



the number K being thus a criterion of the possible maintenance of 

 sinuous or eddying motion. 



The experiments also showed that K was equally a criterion of 

 the law of the resistance to be overcome which changes from a 



* ' Phil. Trans.,' 1883, Part III, p. 935. 



