Axis of Symmetry 



Figure 227 — Relations in axisymmetric flow. 

 See Section 136. 



the lower signs are to be taken. The equations are easily obtained from the definition of ipi 

 as given in Section 16. For example, the flow between two circles drawn with the axis of 

 symmetry as axis and through two points (A, /i) and (A + SA, /x) is 2n-S0 = - 2T7a)ds^q\ insertion 

 of Sip = 5kdip/d\ and division by ^tto^s^ gives Equation [136]]. 



If a velocity potential <^ also exists, it follows by comparison of Equations [136i,j] with 

 Equation [136e] and its analog for q that 



Sk dc^ ^ 1 5/i dip 

 Ss^ ^A "cj 8s d[L 



8y. dcf) 

 8s^ dn 



1 SA dip 

 CO Ss-, d\ 



[136k, 1] 



The signs are explained under Equations [136i,j]. Because of the symmetry, the third term 

 in the Laplace Equation [136g], in which now v = co, disappears. A corresponding equation 

 for is obtained by substituting for d4>/d\ and d(p/dii from Equations [I36k, 1] in the identity 

 d^cp/dixdX = d^cp/dXdfi; 



S^ dip 



S\ 8s dn 



[136m] 



345 



