604 



131 / 3Z?„ 1 \ 



X = -77- I 1 — ~r ^ff TT ~\~ terms of the order Z)„ . . . . . 

 ^0 V 4 X», J 



(58) 



The results of (57) and (58) are substituted into equation (53), 

 and the maximum value of is determined. This maximum occurs if : 



262 / 



98 R 



1 la — , 



R ^ 262 



Finally equation (54) gives: 



2 11 1 



C = 0,0108 -f ^ Ig R + terms of the order -7; ') • • ■■ i^^) 

 n a 



Discussion. 



In this case loo the quadratic law of resistance is asymptotically 

 arrived at (for values of R surpassing 100000 the logarithmic term 

 is little more than 2°/„ of the constant term). Just like what occurred 

 in liie more simple case tlie value of the coefficient C is too high. 

 For channels with smooth walls von Mises gives that (,' ranges from 

 0,006 to 0,0024 if R ranges from 10000 to the greatest values 

 obtained; the formula derived by von KaRMaN's theory gives: 



C=c&. 0,07 R 'U 



For channels with rough walls the dependance of the coefficient 

 C on the value of R is usually very small, so that a quadratic 

 resistance formula can be used, the value of C depending, however, 

 on the dimensions of the irregularities of the walls as compared to 

 the diameter of the channel. The value of C is much higher than 

 in the case of smooth walls; it may even surpass that given by 

 (59). So Gibson mentions values ranging to 0,015 for old cast iron 

 tubes or channels, lightly tuberculated '). 



Laboratorium voor Aero- en Hydrodynamica der T. H. 



Delft, May 1923. 



') The constant term of in this formula has a value of 8 times that of 

 formula (47) An elementary but superficial comparison of the magnitude of the 

 frictional forces exerted on the walls in both cases leads to the same result. 



') R. VON Mises, I.e. p. 63, in connection with the definition of r, given at 

 p. 83/84. In the case of a channel of infinite depth as the one treated here, r is 

 equal to h. 



A. H. Gibson, Hydraulics and its applications (1919), p. 209 (in the formula 

 mentioned at p. 206 is m is 4 time the quantity r introduced by von Mises; 

 comp. Gibson, I.e. p. 194). 



Comp. also L. Schiller, ZS. fur angew. Math. u. Mechanik, 3, p. 2, 1923. 

 and others. 



