Desai and Lieber 



flow. Thus the condition not satisfied by the base flow velocities Ug and Vg will 

 be satisfied by the first iteration velocities Uj and v,. Let, for n > 2, u^ = o 

 and Vn = at both boundaries. From Eqs. (1.60), (1.61), and (1.62) we have by 

 differentiation 



Uj(r,0,t) = + 2_, sin n5 + cos n( 



v^(r,e,t) = 2_, [An(r,t) cos n5 + B;(r,t) sinnS] ; 



^ r^ = 1 



A nC„(r,t) nD„(r,t) 

 u^Cr.^.t) = + } sin nS* + cos n( 



v^Cr.^.t) = 2_, [Cn(r,t) cos n0 + D_:^(r,t) sinn^] ; 



^ n = l 



UgCr.^.t) = + /_. sin ne" + cos ni; 



(1.63) 



(1.64) 



(1.65) 



' E'( r, t) ^ 



v^(T,0,t) = '- ^ [En(r, t) cos n<9 + F;;(r, t) sin n^] ; 



n =1 



where prime denotes partial differentiation with respect to r. 



Consider, first, i/zq, uq, and Vq as given by i/-', u', and v' in Eq. (1.45). 

 Since v = o at r = i, we must have for this case 



V, I = -V - -2 sin , 



•' ' M r = 1 I r = 1 



and 



Applying Eqs. (1.66) to Eqs. (1.63), (1.64), and (1.65), we find that 



A^(l,t) 3 , B„(l,t) = n = 1, 2, ... 

 A;(l,t) = n = 0, 1, 2, . . . 



(1.66) 



(1.67a) 



B;(l,t) = +2 , B;(l,t) = n = 2, 3, . . . ; 



546 



