136 HYDRAULICS AND ITS APPLICATIONS 



If the scale ratio for any two orifices, i.e., the ratio of any 

 two corresponding linear dimensions, is S, the ratio of the areas of 

 corresponding elements of the orifices will be S 2 , while if similarly 

 situated with respect to the water surface, their depths are proportional 

 to 8. 



That the value of C may be expected to be very approximately 

 the same for two such similar orifices was shown by Professor 

 James Thomson 1 in a discussion of which the following is a brief 

 resume. 



Consider two similar and similarly situated particles of fluid (1) 

 and (2). The masses of these and their weights are in the ratio of their 

 volumes, so that we have w\ = S 3 ?r 2 . 



Now if these particles are to trace out paths which shall be similar, all 

 corresponding forces acting upon them must have the same ratio, and it 

 follows that this ratio must be the ratio of their weights since this is 

 fixed. Apart from the forces due to the statical pressure corresponding 

 to the depth of the particle, which are proportional to its depth 

 multiplied by its area, and therefore follow the required law, the only 

 other forces are those due to centrifugal action and to the effect of 

 viscosity. Neglecting the latter, 2 and terming F the centrifugal force, 

 we have 



But if the paths of the particles are similar, i\ = S r 2 , 



. tt - ( LV 



F\- ' UJ 



It follows as a necessary condition that f -- J = S and this condition 



will evidently be fulfilled if ( V = p, Le. f if v* = 2 q h, and if all 



V v 2 ) l>i 



corresponding particles are similarly situated with respect to the free 

 surface. In such a case, the paths of all corresponding particles being 

 similar, the contractions of the two jets will also be similar and the 

 values of C will become identical. A simple ratio thus connects the 

 relative discharge of the two orifices. 



1 " British Association Report," 1876, p. 243. 



2 Since the force introduced by viscosity = u ~ per unit area, we have, if P is this force, 



(I X 



a (5)2 a ( ^ )2 V ~^ T if Vl = v ^**' so that * a S)S - Viscosit y 



therefore tends to prevent exact similarity of flow, unless <* (S) 



