VISCOUS FLOW 69 



Integrating, we get 



Since the velocity at the axis, where r = o, cannot be infinite and since 

 log o = inf., A = o. 



Determining B, so that u = o when r = a, i.e., for no slip at the 

 boundaries, gives 



*-&&{* 1 < 2 > 



So that the flux through the pipe, which equals 



' = -7 . 7 cubic feet per second. 

 8 ju, d # 



Writing -~ as ^* , ^ 2 where ^i and ^ 2 are the pressure intensities at 

 a distance I apart, along the axis of the pipe, we have 



4 



Q = ^ . ^ l ^ 2 cubic feet per second. 



O /U, 6 



The maximum velocity is obtained by putting r = o in equation (2) 

 and equals ^ ~. a 2 feet per second. 



The mean velocity = ^ r . = ^ ^ . c? feet per second. (3) 



area of section 8 //. I 



.'. Maximum velocity = 2 (mean velocity). 



On equating expressions (2) and (3) it is 



readily shown that the filament of mean velocity ^ JIBUt v 



is found at a radius '707 a. ~~^~ 



From equation (2) we see that the curve of Fl( , 33 



velocities across a diameter is a parabola, and 



that the surface of velocities for the pipe will be a paraboloid of revolu- 

 tion (Fig. 33). 1 



If v = mean velocity, we have pi p% = 1 2 Poiseuille's form 



of the equation. (Arts. 63 and 64.) 



The shear stress at the boundary is given by 



d u r dp a p\ p% /^\ 



1 For a curve showing the variations in velocity for stream line flow through a two-inch 

 pipe, see a paper by Morrow, "Proc. Roy. Soc.," vol. 76, 1905. 



