studies on the Motion of Viscous Flows— III 



Let us consider Eqs. (2.29) to (2.33) inclusive, with the stipulation that 

 An ( r , t ) =0 for all n, B^{ r , t ) =0 for n > 3, and Bn(r, t) for n = 1 and 2 

 are the solutions to Eqs. (2.48) and (2.49), satisfying the conditions of Eqs. (2.11) 

 and (2.12), etc. In Eqs. (2.29), (2.30), and (2.31) all the terms which involve D^ 

 and their derivatives drop out, thus leaving them as equations in c^ and their 

 derivatives. On the other hand, in Eqs. (2.32) and (2.33) all the terms involving 

 Cn and their derivatives drop out, thus leaving them as equations in D^ and their 

 derivatives only. The essential effect is that the explicit connections between 

 Cn and D^ are severed. These equations, then, deal with only one set of func- 

 tions; either C^ or D^. 



Since Eqs. (2.29), (2.30), and (2.31) are a set of simultaneous linear- 

 differential equations with variable coefficients, they have unique solutions, if 

 they exist, satisfying the conditions of Eqs. (2.34) and (2.35) with a suitable ini- 

 tial condition when the above stipulations are taken into account. The trivial 

 solutions Cn(r,t) = satisfy these equations and the required conditions of 

 Eqs. (2.34) and (2.35). Hence, if the initial flow conditions are such that they 

 represent a symmetric flow pattern, or if we are considering a steady flow prob- 

 lem, then it follows that Cr,( r , t) = are the only solutions to these equations 

 and conditions. 



Now we made another simplifying assumption similar to as follows: 



D^(r, t) = n = 3, 4, . . . . (2.51) 



As a result, we obtain the following equations governing the functions D^ and 



? (l + ^ - ^0 ®2 ' ^ ^°2^^ " ^i*2)+ 77 (^2^ - Di«;) 



+ — (B^S^ - BjS;) + — (B^S; - B;^^) = Re ^ ; (2.52) 



3); . i ®; - ± ®, . ^ (i - -1 - ^) ®; - ^ (i . ^ - b;U, 



2 r 2 J.2 2 2 \ r2 W 1 2r \ r^ ^j^ 



+ i^(D;55j-DjS;) + :^ (b;»j-Bi«;) = Re^ ; : (2.53) 



(r- i- B,) ®^- 2 (l + -^- B;) 5), + (2),Bi-2B23);) 



+ (!BjD^- D^!B^) + 2 (S^d; - D^a;) + (\B'^-B^^) + 2 (SjBj'-BjS;) = 0; (2.54) 

 (B^iD^-B^Sj) + (D2»2- 1^2*2) + (JBjB^-S^B^) = . (2.55) 



569 



