Theory of Turbulent Flow Between Parallel Plates 

 ^20 ~ "JQ'^^i cos ax - \ ( 1 - a) 0/3 cos ax + /3-^dC cos dx 



+ jbD cosh bx 

 V^g = -aaA^ sin ax + j ( 1 - cr) aawBj sin ax + ja/S^Cj sin dx 



- aD sinh bx 

 X - SaA sin ax 

 T - cT(7-l)feagB sin ax + ctC sin dx 



+ (y - 1 ) ojD ^ sinh bx 

 P= JQ(7~^) Sag/3 ^coB sin ax- j(l-cr- aq)/3^(oC sin dx 



+ coT) sinh bx 



7 - 1 



54,.,3 



a2 % -jo) - X , d2 % -jcro)- \ , b2 % \ - /32a)2 + jS/32g + j M + I__L + q 1 /3 



These solutions are essentially the same as the previous core solutions. 

 In the boundary layer (q?' = ±N, x = ±1) 



[D2 + a2]U + qD [DU + j ^/a V + jSX] - DP = , ' 



[D2 + a2] V + jaq [DU + j A.''a V + j&X] - jaP = ' . ', "'' 



[D2+ a2] X = +gU " ' ' " '■ • 



[DU+ jVa V+ jSX] - 7/32gX - yS ^-^V + 7/32ja;P - /32j^T = - .- ', ., ., . 



a ■ -. ■ 



^ [D2 + a2+ j (1 -CT)aOT + 2ejSgU- g[+2eD+7 -1]X- ■^g[±2eD+ y - 1]V + (y - I) y^P = . 



This is the same as Eqs. (9). The solutions were previously written in 

 terms of exponentials as independent boundary -layer solutions. 



The first solution set should be transformable into the second solution set 

 by perturbation. The present purpose would be to relate the constants for the 

 core solution to the boundary-layer solutions, and second to derive, if possible, 

 an explicit perturbation theory. 



In accord with [10], we can surmise that a convergent perturbation is of 

 the form 



V = Vp cos F' + Vj sin F' + V^ cos F' , etc. 



an in-phase perturbation (V2 cos — where Vj is even when Vq is even) and an out 

 of phase perturbation (Vj sin — where Vj is odd when Vg is even). The functions 

 Vj, V2, F' will depend on qp and its derivatives, to first order in a first-order 

 theory. (While the decomposition into F', Vj and V^ is not unique F' is chosen 

 for convenience, e.g., if 



743 



