Desai and Lieber 



Let us put, for brevity, 



0l(r,5,t) 



M+ Bj(r,t) + Dj(r,t) + Fj(r,t) + ••• sin 6 ; (2.39) 



^^(T,e,t) = [B2(r,t) + D2(r,t) + F^Cr^t) + •••] sin 2^ ; (2.40) 



0^(r,^,t) = [B^(r,t) + D„(r,t) + F„(r,t) + •••] sinn0 . n>3 (2.41) 



Then we may rewrite 



0(r,0,t) = ;/;^ + 0^ + 0^ + ... (2.42) 



From Eq. (2.39) we see that the terms in sin 0, due to various iterations, 

 combine with the corresponding term in the potential stream function i//q. To- 

 gether they make it possible for the stream function 4) and its derivatives to 

 satisfy the required conditions at the cylinder wall. This is achieved by a 

 modification of the potential flow field. K the lines ^^ = constant are plotted 

 they will have two axes of symmetry, viz., (a) the polar axis and (b) the line 

 at right angles to the polar axis at the origin. The polar axis itself is one of 

 the lines ^^ = constant, but the other axis is perpendicular to all the lines ^j = 

 constant. Evidently the lines i/^j = constant represent a streamline pattern 

 markedly similar in structure to the potential streamline pattern. Similarly, if 

 we plot the lines ^2 ~ constant, they will have four axes of symmetry, two of 

 which are the same as for 0j = constant. The other two axes of symmetry are 

 mutually perpendicular lines making an angle of 45° with the polar axis. In 

 this case, the polar axis as well as the line at right angles to it at the origin are 

 the lines 4^2 - constant, while the other two lines are perpendicular to all the 

 lines 02 = constant. K we write 



u , = — 



1 ^^0 1 



cos 6 



' ' r2 



+ Bj + DJ + 



(2.43) 



2 r 



^0, 



- (B2 + D2 + • • •) 2 cos 2( 



•(B' + D' + • •• ) sin 26* , 



B, ^ . ^ (2.44) 



where prime denotes partial differentiation with respect to r, then we have 



Vj(^) = +Vj(77-^) ; (2.45) 



566 



