FREE VORTEX 



97 



i) v 

 .'. in any horizontal plane .-^ + ^ = constant. 



Differentiating, 



JL J'J + ^ d v - o 

 W ' d r T i ' dr~ 



d) 



Introducing the condition for rise in pressure across a stream tube, 



dp }Vv* ,,. , 

 v\z.,~~- = ^-9 this becomes 



f a , v d v 



dr 



gr 



9 r fl <l '' 



d r , v d r d v 



.'. -y + - = 0, or --- = . 

 d r r r r 



Integrating we get log e r + log e v = constant = B. 

 v r = constant = e B = BI. 



(2) 



i.e., in a free vortex the velocity varies inversely as the distance from the 

 axis of rotation. 



It follows that the increase 



of pressure with radius is 



identical with that in inward 

 radial flow. 



Thus if pi, vi, ri, are the 

 attributes of a point in the 

 same horizontal plane as p, 



r, r, 



2 c 2 ^ 



P Pi ?'l I -, ?'l /Q\ 



w~ ~~ o~^ 1 L {* FIG. 50. Free Vortex. 



** y \. ' 2 / 



Putting -^ constant in Bernoulli's equation we get the equation to the 

 curve of equal pressure, that is ^ |- z = constant = C, and substituting 

 for v in terms of r from (2) we have 



. n _ z - 01 



the equation to a hyperbolic curve of the nature y x 2 = A, and which is 

 asymptotic to the axis of rotation and to the horizontal through z = C 

 (Fig. 50). 



A Free Spiral Vortex may be considered as a case of cylindrical vortex 



H.A. H 



