A. S. Iberall 

 X = -1 



U = -^^C^Dj e 



V = j acoT) e "* 



Jgc, 



T^3- 



■j(7- 1) 



Jg-'c, 



1 + j 



\ - C 2 

 4 



q - 



4e 



7 - 1 



2ejg^c/ 



(7- 1) 0)3 



■}u>^ 



^ - c 2 jqogc 

 1 + ( 1 + q) r— ^ + ^ 



D,e 



Note: Building these solutions requires some a priori estimate of relative 

 magnitudes of various parameters, typically the following: 



parameters large compared to unity— -.>, g, R^^, M^, v^ (It is not 

 likely that ^ is less than 100.), 



parameters small compared to unity — /3 , fi'^aj, /32g, (i'^\^, 



parameters that are quite bounded— ^, i>, a, SRqo/w, /3g, /3m (It is 

 not likely that /Sco is greater than 100.]. 



However, there are some parameters, such as w/g, whose bounded magni- 

 tude is uncertain. Also, in developing coefficients in series, while use can be 

 made of the small magnitude of /3, /3^oj, /3^g to permit rapid cutoff of such se- 

 quences, t his m ay not be done with regard to the square root of such magnitudes, 

 e.g., y/i3, 7/32ajj 1/y/^. Thus, sequences must be carried forward at least to such 

 terms, i.e., all series must be imagined in terms of such half-power expansions.] 



SECULAR EQUATION -SELF-GENERATED PROPAGATION 



The vanishing of the secular determinant emerges from satisfying boundary 

 conditions u = v = w = x = 3" = 0, atx = ±l with these primitives. 



A preliminary result can illustrate how limit cycles are generated. A secu- 

 lar determinant may be obtained first from the core solutions, which assesses 

 the nonlinear contribution of a substantial mean velocity (Reynolds number) over 



720 



