Self-Conveoting Flows 



J KE 

 d 



£-= - W:^ • CM (17) 



dt 



where for large values of \\i, Cp must approach sorae limiting value 

 Cp*, while for small values of i|j (ijj -^ 1) , C^ "^ Cp(l) » (We/Wj)^. 



Laws of Motion. Limiting Cases; Weak Circulation {^ "*' i) . 

 In this case the inner and outer flows are almost matched and the 

 deviation from the ideal vortex motion is small. The laws of motion 

 in their appropriate form become, 



Volume . 



^ = (q> - 1) • «'(!) (5a) 



Vorticity, 



.^i^^ = - A,a'(l) . (^ - i)^ . W, (10a) 



dz --I \ / VT / "e 



Momentum. 



K„(l) • WgR^ = const. (Ua) 



Energy. 



2_2 



K('>%^=-(1-)w/r-W (IVa) 



Combining (5a) and (10a) leads to the result, 



j^l*A,(V.-l) 



We « -TTfTTTTTT A, ^ (18) 



Whereas, (11a) requires 



W. « -^ or z « t'/' (19) 



® R^ 



so that the presumed motion can take place only if A| = or 



A| = (i|j - 1)"'. Combining (17a) and (10a) leads to the requirement 



that the velocity ratio be constant and have the value 



333 



