410 Mr. L. K. Wilberforce on the Calculation of the 



the form in which the necessary corrections must present 

 themselves may be of service. 



Osborne Eeynolds (Phil. Trans. 1883) has experimentally 

 shown that, in a tube of given diameter D, if the mean ve- 

 locity is not greater than a certain critical value the motion 

 of a fluid at points not very near the ends (t. e. about 120 dia- 

 meters off, in his experiments) settles down into the state 

 which we have hitherto assumed to exist throughout. He 

 refers to these experiments in a later paper (Phil. Trans. 1886), 



and there gives — — = 1400, /jl being the viscosity of the 



liquid, as an equation from which to determine the critical 

 velocity ; but from the experimental results given in the 

 earlier paper it would appear that the number should be 2000. 



It follows that if the tube be long compared with its dia- 

 meter, and if the mean velocity be below its critical value, 

 both of which conditions we shall assume, the motion near 

 each end of the tube both within it and outside will depend 

 only upon the mean velocity and not upon the length of the 

 tube. 



We shall suppose that the fluid flows from a vessel into the 

 tube, and from the tube into a mass of fluid of the same kind 

 in another vessel ; and that when we investigate the effect of 

 a change in the diameter of the tube, we change the dimen- 

 sions of the vessels in the same proportion, and so arrange the 

 tube with respect to them that the whole system shall remain 

 similar to itself, the length of the tube alone excepted; then, 

 from the form of the equations of motion of a viscous fluid, we 

 deduce that the character of the motion at similar points in 



the different experiments will be a function of — - only. 



A 6 

 We may consider that this motion of unknown character 

 exists in the vessels and in the tubes up to distances of 



J)xfi ( VP ) and D x/ 2 ( — — ) from the ends, and that in the 



rest of the tube the distribution of velocity is the normal one. 

 Since for the case of similar fluid systems moving similarly 

 the dissipation function is proportional to volume x coefficient 

 of viscosity x {gradient of velocity) 2 , we have for the energy 

 dissipated per unit time in the vessels and ends of the tube, 



The rate at which energy is dissipated in the rest of the tube 



