182 



HYDRAULICS AND ITS APPLICATIONS 



experiments were made to determine how such a variation affected the 

 resistance. 



A possible law of such variation may be deduced from purely theoretical 

 considerations, the only assumptions made being that the resistance of 

 each element of the rotating surface is proportional to the same power 

 n of its velocity, and to some power of the viscosity //, and of the density 

 w of the fluid, both of which vary with temperature. 1 



On this assumption it may be shown that the resistance, other things 

 being equal, is, as in the case of pipe flow, probably proportional to 

 fj?~ n w n ~ l , where n is that power of the velocity to which the resistance 

 at constant temperature is proportional. 



For the purpose of determining the temperature variation three sets of 

 experiments were carried out, using respectively a 12-inch brass disc in 

 a machine-ironed casing, with 1^ inch clearance ; the same disc in a 

 rough cast-iron casing, with J inch clearance ; and a 12-inch disc with 

 radial vanes J inch deep and with f inch side clearance, in a painted 

 and varnished casing. The values of n were determined in each case 

 from a second set of experiments carried out as nearly as possible at 

 constant temperature, all the results being corrected to 65 F. by an 

 application of the foregoing hypothetical formula. This involves the 

 method of successive approximation for finding the true value of n, but as 

 the temperature corrections were small (never above about 2J per cent., and 

 generally very much less) the first approximation, differing as it did from 

 the true value by not more than 1 per cent., was in general sufficiently 

 accurate. In this way the following values were obtained : 



Series A. 

 12-inch brass disc in smooth casing . . . . . 1 inch clearance, n = 1*785 



Series B. 

 12-inch rough cast-iron casing . . , n = 1-800 



Series C. 

 12-inch ,, with radial vanes f , n = 1-950 



The results of the temperature-variation experiments are then as 

 follows : 



SERIES A. 1,400 revolutions per minute. 



By the theory of dimensions. The method is applied to pipe flow on p. 198. 



