Self -Conv eating Flows 





, _^^ or W,/W,^«(:^) (22) 



R' 



and, substituting (22) and (5b) into (lib) leads to the differential 

 relation. 





which has the solution, 



where 



and 



£L K, . z = \- + IS? R + const. (A| ^t 0) (24) 





— K. z = in R + ^R + const. (A, = 0) 



or 



£!_ K, 1LJ_Eo) = i n R + ^ (R - 1) (24a) 



Substituting dR/dz derived from (23) into (5b) yields a relation 

 between ijj and R , 



^= ^T^^ <25) 



and, finally, it may be shown that, 



w ..-2 k- -C + A,) 







The type of motions which ensue from this theory in the case 

 of strong circulation are seen to depend very much on the value of 



335 



