340 



HYDKAULICS AND ITS APPLICATIONS 



which influence this distribution, that, as might be expected from the 

 nature of the case, the formulae so far collected can only be considered 

 as giving useful approximations to the required result, and this is more 

 particularly the case where the flow in a natural channel of irregular 

 section is under consideration. 



By making one or two assumptions as to the circumstances governing 

 the flow in an open channel, a theoretical formula may be deduced, which, 

 while only applying so far as these assumptions are justified, may still 

 serve as the rational basis of a more exact empirical formula, for giving 



the distribution of velocity. Such a for- 

 mula will now be considered. 



Suppose the stream to be sensibly paral- 

 lel ; of width which is great in compari- 

 son with its depth ; flowing steadily ; and 

 that the resistance to flow is due entirely 

 to simple viscous shear, a state of affairs 

 never exactly realised in practice. 



Let y be the vertical distance from the 

 surface, of a stratum of the fluid, 8 y the thickness, 8 I the length, and b 

 the breadth of the stratum (Fig. 152). 



The weight of this element of fluid = W b 8 y 8 I. 



The resolved part of this weight ) TTr , * 5. 7 . 

 . r \ = Wb 8 y8 I.sm0. 



in the direction of motion j 



The difference of tractive force on the upper and lower faces of the 



d? v 

 stratum = p ^3 . b . 8 1 . 8 y, where ^ is the coefficient of viscosity (p. 67) 



and where v is the velocity of flow in the direction of the stream. The 

 pressures on the two ends of the stratum are equal since these are at the 

 same depth and are of the same area. Also since the stream is wide, 

 the variation of shear on the two vertical sides of the stratum may be 

 neglected, as explained on p. 66. 



/. W b 8 y 8 I sin = /ot ~~ . b 8 I . B y 

 W sin d* v 



FIG. 152. 



the negative sign denoting that the resultant shear force acting upon the 

 element is in opposite direction to the force W sin B. 

 Integrating this expression twice, we get 



(1) 



