Desai and Lieber 



The boundary conditions in Eqs. (1.77), (1.78), (1.83), and (1.84), when ap- 

 plied to the expressions of Eqs. (1.63) which result from the periodicity condi- 

 tion on u^ and Vj, imply that the following conditions on the fvinctions A^ and 

 B„ should hold: 



For a steady flow, these boundary conditions together with the differential 

 Eqs. (2.3), (2.19), and (2.8), suffice to determine the functions A^ and B„. For 

 a time-dependent flow, the nature of initial conditions also needs to be critically 

 considered. Once the functions A^ and B^ are obtained uniquely, they deter- 

 mine uniquely the stream function ^i and hence the first iteration velocity field 

 u ^ and V J . 



In view of the harmonic structure of the stream function, the pressure field 

 also has the same structure, which is described in detail in Refs. 1 and 2. 



SECOND ITERATION 



To obtain u^, v^, and p^ which satisfy Eqs. (1.51), (1.52), and (1.53) to- 

 gether with the boundary conditions of Eqs. (1.79), (1.80), (1.85), and (1.86), we 

 will solve, as in the case of the first iteration, Eq. (1.54) together with the con- 

 ditions in Eqs. (1.79), (1.80), (1.85), and (1.86) to obtain first 02 a-^d then u^ 

 and Vj. Since ^^t ^i^^ "Pn ^^ ^ periodic function of 6 and has a Fourier series 

 representation, an examination of the terms in Eq. (1.54) reveals that we must 

 here obtain a Fourier representation of terms involving the products of two 

 Fourier series. For this purpose, we refer to the following two theorems. 



Theorem II (Perseval's Theorem) [41]: If f (x) and F(x) are 

 square integrable functions defined on (-tt, tt), for which 



v^ 



f(x) = — + ) (^n COS nx + b^ sin nx) 



" n= 1 



558 



