46 Transactions. 



consequence the efforts of experimenters to discover the nature of the 

 relationship has been fruitless, and, as little or nothing is known of the 

 transformation of energy within fluids in sinuous motion, a precise mathe- 

 matical solution was impossible. It may, however, be deduced from 



certain dynamical principles that the resistance is some function of ( — J 



\vdj 



provided that there is a proportionality between the dimensions of the 



eddies and of the cross-section of the pipe, leaving the form of the 



function to be determined by experiment. 



The law of resistance, then, in its most general form, which applies to 



both states of motion, is expressed as follows : — 



3-*(a) < 2 > 



where <£ stands for " function of" and the other symbols have the same 

 significance as in equation (1). As already explained, the relation 

 between the quantities in the linear state is a simple one, the left-hand 

 expression in equation (2) being a simple linear function of the right hand. 

 As regards sinuous or turbulent flow, it was supposed at one time 

 that the nature of the function was of the form 



rs 



/>i2 



•U" < 3 > 



but it is now known that this form is defective, and that the range of 

 observations upon which it was based was not wide enough to determine 

 the true form ; it was soon found that equation (3) did not fit the facts, 

 and in consequence a modification of this was adopted in which v was 

 treated as a constant, and independent indices given to v and d, yielding 

 a formula of the form 



v = kr*sv (4) 



where k, x are constants and r is the hydraulic mean depth numerically 

 equal to d/4 for round pipe. 



This formula is one of considerable flexibility, and of late the whole 

 phenomenon of the flow of water in pipes has been analysed afresh 

 and expressed in the form given in equation (4). Its adoption has not, 

 however, contributed anything towards extending our knowledge of the 

 subject, and it is much to be regretted that steps were not taken to 

 extend the range of observations when equation (3) was found to be 

 defective. Tnis aspect of the question has been apparently overlooked. 



Such a series of observations extending over a wide range was recently 

 conducted in the National Physical Laboratory by Stanton and PannelP 

 upon oil, air, and water in smooth brass pipes. The diameters of the 

 pipes used varied from 142 in. to 5 in., and the mean speed from 

 a fraction of a foot to 20 ft. per second. These combined with other 

 observations upon the flow of water in smooth pipes when plotted 



with -- as ordinates and log. — as abscissae were found to be sufficientlv 

 v % V 



near to enable a curve to be drawn through the mean which was fairly 



representative of the whole, despite the fact that the condition of 



geometric similarity was not observed in respect to the surface of the 



* T. E. Stanton and I. R. Pannell, Similarity of Motion in Relation to the 

 Surface Friction of Fluids, Phil. Trans Roy. Soc, A.'vol. 214, pp. 199-224, 1914. 



