Theory of Turbulent Flow Between Parallel Plates 



the central region of the tube. It does not assess the contribution of curvature 

 of the mean velocity field near the wall. Letting the dual set of even and odd 

 core solutions vanish at x = ±1 leads to the following conditions: 



(both even and odd solutions for the vorticity vanish independently) 



from which 



d 2 = , 

 4 



k = /32Mg2 ^ /32a;2 , 



and &/^2g is small. • — , .. . ■■•: r* 



The eigenvalues hereby obtained for self-generated propagation are instruc- 

 tive but not necessarily complete or correct. Instead of the vorticity actually 

 vanishing, a weak generation of vorticity may develop. Most interesting is the 

 expectation that the "mean" propagation is likely to be an elastic wave ( a^ + 



Returning to satisfying boundary conditions with the wall solutions, it can 

 be shown that the leading- order terms for these solutions are the following: 



U = co(l - a) 



2ECr/32gc 



( 1 - ct) y/Joj 



V = -aco[(l- a) S - 2ea-/32gaJ Ae" 



X = w [(1- a)\- 2eo-S/32gaj] Ae"^''!" 



± ja)(l - CT) /]Z Be-(^e/'^)x g-'^a- 



Be±(«B/2-)Xe-'^2'^ + j(l-a)/32a;a re"'^3'^ ±a;c ne"''^' 



+ j acoBe 



o- (1- CT) coCe ^"3" 



-j (y- l)t^2De±c4x 



Invoking the conditions for a nonzero set of coefficients A, b, c, d, the 

 vanishing of the determinant 



W ( 1 - Cr ) 



-aw[(l-CT) S-2ea/32gaj] 



oj [(1- a)\-2eo-5/32gaj] ±jw(1-ct)\/j" 

 



leads to the following secular equation 



2eCT/32ga 



(1 - cr) vG^ 



j aco 







= 



721 



