Desai and Lieber 



D„(l,t)=0, D;(l,t) = 0, D„(ht,t) = 0, D;(h*,t) = . 



(2.36) 



n = 1, 2, 3, ... 



Equations (2.29) to (2.33) inclusive, together with the homogeneous boundary 

 conditions of Eqs. (2.34), (2.35), and (2.36), complete the formulation of the sec- 

 ond iteration. 



SIMPLIFYING CONSIDERATIONS 



The sets of subsidiary equations and the boundary conditions in Eqs. (2.3) 

 to (2.13) inclusive, for the first iteration, and Eqs. (2.29) to (2.36) inclusive, for 

 the second iteration, are in their most general form consistent with the assump- 

 tion of the existence of Fourier representations for the stream function i/zj and 

 02. These equations and the boundary conditions can be simplified considerably. 



The terms in cos n0 and sin nd respectively represent an asymmetric and 

 a symmetric flow pattern. If the initial flow conditions are such that they repre- 

 sent a symmetric flow pattern at the time t = 0, or if we are considering a 

 steady flow problem such that A^, B^, C^,, and 0^ are all assumed to be time- 

 independent, then it is evident from Eqs. (2.3), (2.4), and (2.5) together with the 

 boundary conditions of Eqs. (2.9) and (2.10) that An(r, t) = 0, (n = 0, 1, 2, . . .) 

 are admissible trivial solutions to these equations and conditions. Similarly, 

 c„(r,t) = are admissible trivial solutions to Eqs. (2.29), (2.30), and (2.33) 

 together with the conditions of Eqs. (2.34) and (2.35). Since the set of Eqs. (2.3), 

 (2.4), and (2.5) is a set of simultaneous linear differential equations with vari- 

 able coefficients, they have unique solutions, if they exist, satisfying the condi- 

 tions of Eqs. (2.9) and (2.10) and suitable initial conditions. The same is true 

 of the set of Eqs. (2.6), (2.7), and (2.8) with the conditions of Eqs. (2.11), (2.12), 

 (2.13), and suitable conditions for the functions B^,. Hence we can conclude that 

 A„(r,t) = are the only solutions. Then a similar argument shows that the 

 set of Eqs. (2.29) and (2.30) with the conditions of Eqs. (2.35) and (2.36) has 

 only Cn(r,t) s as the only solutions. The only nontrivial solutions are in B^, 

 and T)^. And these correspond to symmetric terms in 0j and ^p^ respectively. 



Since the equations for >//„, n > 3 are structurally similar to the equations 

 for 0j and i^j, and the conditions on the asymmetric parts of these stream 

 functions (//„ are the same as those on the asymmetric parts of the stream 

 functions ^^ and <~p2) we can conclude by induction that the solutions corre- 

 sponding to the asymmetric parts of the stream functions 4>^ must be only the 

 trivial ones. This leads to the following theorem. 



Theorem IV (Symmetry Theorem): If the initial flow conditions 

 are such that they represent a symmetric flow pattern at time t = 

 0, or if the flow is steady, then the resulting flow pattern must be 

 symmetric about the polar axis for all time t. Moreover, an 

 asymmetric flow pattern must be time-dependent and can result 

 only if an external disturbance at any time t or the initial flow 

 conditions at time t = introduce an asymmetry. However, a 

 time-dependent flow is not necessarily asymmetric. 



564 



