28 



z(t) = z (t) + Az,(t) + A-^z,(t) +. 

 1 i- 



X + Aoi + A^ao +. 



(21a) 

 (21b) 



Sxobstituting these two expansions into (20) and 

 (18) and equating the terms of equal order in A 

 yields the set 



2ir/fi periodic in time. All of the equations of the 

 set (23) , (24) etc. have the form 



dz. 



— f - (D-A I)z. 

 dt P D 



h(t) 



(28) 



where h{t)is a periodic vector function which has a 

 Fourier series representation of the form 



dz 

 o_ 



dt 



(D-X I)z 

 P o 



(22) 



h(t) 



c 



"l,' 



ikflt 



(29) 



etc. 





dt 



(D-Apl)z„ 



(E-a,I)z (23) 



1 o 



(E-a I)z - a z (24) 

 1 1 2 



where hj. are constant vectors and the p-th component. 



i^op °f K 



is zero. (This property is enforced by 

 the solution procedures.) 



Then, application of the Fredholm Alternative 

 for solving (28) yields the requirement that 



<h(t) ,y> 







(30) 



Note that the constant coefficient matrix of 

 these equations is 



where y-j 



X I 

 P 



\'' 



r^, 



1, . 



N J 

 N' = N 



4. 



Ascuming condition (30) to hold (this will be 

 achieved by properly selecting the a^'s), the 

 general solution, z^, can be written as 



z. = exp[(D-A I) t] X 

 3 P 



The only 2iT/n periodic solution that is possible 



for Eq. (22) when (Aj-Xp)7^mfi for m = 0,1,2,. 

 j 7*= p is the solution 



. and 



C + I exp[-(D-A I)s]h(s)dsP (31) 



c(6 .) 

 PD 



(25) 



where & ■ is the Kronecker delta and c is an arbi- 

 trary complex constant. This statement is merely 

 a restatement of the fact that the eigenfunction 

 corresponding to the eigenvalue, Ap , is the p-th 

 column of matrix B, i.e., the least stable eigen- 

 mode of the underlying steady Blasius flow. 



Since the solution (20) requires that z(t) be 

 periodic with period 2-n/a in t, we shall need the 

 inner product <f,g> defined by 



- - w r 



2-n/a. 



N' 



E 



j-i 



f .g.*dt. 

 3D 



(26) 



where the asterisk superscript denotes the complex 

 conjugate. We shall also need the adjoint eigen- 

 function of Eq. (22) that corresponds to the eigen- 



value, Yr 



0, and that is 2iT/f2 periodic in t. This 



eigenfunction is 



Equation (31) is easily evaluated because 



Y n t Y ' t 



exp[(D-A I)t] = r^ ,...,e ^ __\ (32) 



Equation (32) is the main reason for diagonalizing 

 the matrix, (1/R|5^) (P+iaJ) . It makes evaluation 

 of the exponential matrix and the integral of Eq. 

 (31) trivial. Thus, by evaluating Eq. (31), and 

 requiring that z-;(t) be unique and 2iT/f! periodic 

 in t, values for the constant vector, fg, are ob- 

 tained which eliminate all the non-2'iT/n periodic 

 functions from (31). The result of these calcula- 

 tions is 



kS. I ikflt 



(33) 



k=-«> 



^0 = '^'^j' • '27) 



For convenience we normalize z and y so that 







<z ,y > = 1 

 ■'o 



by setting c = d = 1. 



The solution of any one of the equations in the 

 set (23) , (24) , etc. is obtained from the solution where 

 of the previous member, by the application of the 

 Fredholm Alternative and the requirement that these 

 solutions are unique, i.e., they do not contain 

 multiples of the eigensolution of Eq. (22) and are 



The solution procedure then is to apply Eqs. (30) 

 and (33) to each of the Eqs. (23), (24) etc. in 

 sequence starting with (23) . These calculations 

 have been programmed and are quite easily performed. 

 (Our program does these calculations to the 7th 

 order term, but more terms can be easily incorpo- 

 rated.) We are mainly interested in the first two 

 perturbation terms which result in 



0; Zj(t) 



:(1) 



-(1) iat , -(-1) -int 



C e + C e 



.(1) 



3P 



in-Y-i 



(34) 



(35a) 



