68 HYDKAULICS AND ITS APPLICATIONS 



If pi p% = fall of pressure between two points at a distance I apart 

 this becomes 



O = - ^ Pl ~ Pz 

 3 H ' I 



The maximum velocity occurs at the axis, where y = 0, and equals 



J- tf *. 

 2 jx ' dx 



flux over a section O 1 72 d p 



The mean velocity = - 7 -3-* 



area of section 2 /t 3 ;x a a; 



Maximum velocity _ 3 

 Mean velocity 2* 

 The shear stress on the boundaries is given by 



du dp h Pl -p 2 

 *d J dx I 



ART. 25. STEADY FLOW THROUGH A CIRCULAR PIPE. 



Suppose the pipe to be horizontal, and, neglecting the effect of gravity, 

 assume the flow to be produced by a uniform difference of pressure head 

 along its length. Let the axis of x be the axis of the tube, and let a be 

 its radius. Using the same notation as in the preceding example, and 

 assuming the velocity everywhere parallel to X, we have v = o, w = o. 



.'. -~ = ; -~ = 0, as in the previous case. 

 dy d z 



The tangential stress, or tractive force per unit area, on a plane per- 

 pendicular to a radius = jw, y-. 



Hence for a cylindrical shell concentric with the pipe, of length Bx and 

 having inner and outer radii r and r -}-or, the difference of the tangential 



force on the inner and outer surfaces will be -= | 2 IT r . 5 x . /ut -= I & r, 



d r ( d r \ 



and this must be balanced by the difference of normal pressure on the 

 two ends of the shell. 



Since -J^ = o, the pressure intensity p at any point of the section is 

 uniform. 



dp* d ( n d u } * 



r .-j^- . & x = -,- \ 2 TT r . S x . /x -3 } 8 r 

 d x d r [ ^ d r j 



j r c u -P- - 



dr ( d r ) ~~ d x ' M' 



