- Desai and Lieber 



These considerations show that the terms in sin 2d are the ones that account 

 for the structure of the flow field in the wake of the cylinder. We conclude there- 

 fore that these terms in sin e and sin 2e which portray the essential features of 

 the flow field and which completely account for the drag on the cylinder are the 

 most important terms in the Fourier representations of the stream functions ^ 

 and 4j^, n = 1, 2, 3, ... . With this in mind, let us introduce the following sim- 

 plifying assumption. Let 



B^(r,t) ^ , n = 3, 4 (2.47) 



Then Eq. (2.8) for n = 4, 5, 6, . . . and the conditions of Eqs. (2.13) are com- 

 pletely satisfied. The equations for n = 1, 2, and 3 are the following: 



K ^ i S; - ^^ fi, . ^ (l ^ ^)% . |(l - -^)S- Re '-^ ; (2.48) 



S; . 1 S^ - -1 «, - ^ (l . J-) S, . 5£ fi _ J_\ S; . Re '-^ ; (2.49) 



-^(l^^V^.^^fl- -^^^^^O . (2.50) 



. .^J ' 2 \ r^j ' 



Equations (2.48), (2.49), and (2.50) are a set of three equations for the two 

 unknown functions Bj and B2. We observe that the Eq. (2.50) is a first order 

 equation in Bj, while Eqs. (2.48) and (2.49) are second order equations in S^ 

 and 'B2 respectively. The set of Eqs. (2.50) and (2.51) together with the boundary 

 conditions of Eqs. (2.11) and (2.12) form a well-defined boundary value problem 

 when adjoined with a suitable initial condition, if the flow is considered time- 

 dependent. If we solve this set to obtain unique solutions Bj and B^ and then 

 find on substitution of these solutions into Eq. (2.14) that it is not violated to 

 any significant degree in the domain, then we can conclude that these solutions 

 are good approximations to the exact solutions to the set of Eqs. (2.6), (2.7), and 

 (2.8), satisfying the conditions of Eqs. (2.11) to (2.13), inclusive. We therefore 

 decide to verify this a posteriori and proceed to solve Eqs. (2.48) and (2.49), 

 subject to the conditions of Eqs. (2.11) and (2.12), etc. We note here the in- 

 trinsic symmetry and the consequent beauty of these two equations which govern 

 the most significant terms of the stream function xp^. 



If we compare the set of equations for the first iteration with that of the 

 second iteration we see that in the first set the equations governing the functions 

 A^, and their derivatives do not contain the functions B^ and their derivatives 

 and vice versa, whereas in the second set the equations contain C^, D^, and 

 their derivatives all mixed together. This means that the functions A^ and B^ 

 are not connected explicitly. However, they are connected implicitly through 

 the time variable t ; and this connection will be lost when the flow is assumed 

 to be steady. The functions c^, and D^ are connected explicitly. If the body is 

 geometrically asymmetrical about the polar axis, as would be the case if an el- 

 liptical cylinder were placed with its major or minor axis inclined at an angle 

 to the flow direction, then again it can be shown that the resulting equations cor- 

 responding to Eqs. (2.3) to (2.8) connect the functions A^,, B^,, and their deriva- 

 tives explicitly. 



568 



