Desai and Lieber 



As discussed in the case of Eqs. (2.48), (2.49), and (2.50), we regard Eqs. 

 (2.52) and (2.53), which are of second order in 0)j and iDj, respectively as the 

 governing equations for the functions Dj and D2, and consider Eqs. (2.54) and 

 (2.55), which are one order lower than Eqs. (2.52) and (2.53), to represent, in 

 a sense, the error involved in the assumption. 



EQUATIONS AND CONDITIONS FOR STEADY FLOW 



As explained above, we may put A^ ^ c^ = in the expression for the stream 

 functions ^j and ^p2 when the motion is considered to be time-independent. To 

 simplify mathematical analysis, we assume here that B^ = D^ = for 3,4,...= 

 n. This enables us to reduce the stream functions i/'j and v^j ^^ the following: 



'Pj^(t,0) = B^(r) sin 61 + BjCr) sin 25 (2.56) 



02(r,6') = Dj(r) sin ^ + D2(r) sin 20 . (2.57) 



First Iteration 



Governing Equations 



2 ■ r ^--2 ^2 ■'■'2 2r \ -2 r-^i o \ r2 ; 1 



where 



Error Equation 



2 ^ \^ ' 2- 2 



570 



(2.58) 



S; ^ i S; - A s - ^ f 1 + ^ U, + ^ f 1 - A-") »; = ; (2.59) 



!B, = B'; + i b; - -L B^ ; (2.60) 



«2 = B'^ + i B^ - 4 ^2 ■ (2.61) 



Boundary Conditions 



Bj(l) = B^Cl) - B^(l) = , B;(1) = +2; (2.62) 



B^(hf) = B^Cht) . B;(ht) = B^(hf) = . (2.63) 



- i)s; - 2(1 + i-jS^ = . (2.64) 



