Studies on the Motion of Viscous Flows— III 



1 V^ 



^ m=l 



Mjb;,„-b;.j + m(3„(A;,„-A;.„)] 



(2.27) 



In Eq. (1.61) we have 



-AoCr.^.t) 



Cn(r,t) 



^ C^(r,t) cos n0 + D^(r,t) sin n( 



Putting 



1 r^. 



e^Cr^t) - q(r,t) + 7C;(r,t) 



e„(r,t) = C;(r,t) + ^C;(r,t) - ^C^(r,t) 



(2.28) 



3)^(r,t) - D;;(r,t) + }D;(r,t) - -^D^(r,t) , 



and using Eq. (1.61) in Eq. (2.19), we obtain an equation which involves the 

 trigonometric functions cos md and sin mO. 



Since the terms in cos me" (m = 0, 1, 2, . . .) and sin m5 (m = 1,2,.. .) are 

 linearly independent, Eq. (2.19) can be satisfied only if the coefficients of these 

 terms are identically equal to zero. This gives the following equations connect- 

 ing the functions C^ and D^ : 



1 /,3„ 1 ^,\ Re /, 1 V:,, Re 



Re 



E "(Dnd;-c„B;) 



CD 



Y nCS^A'-es') 

 / ■ ^ n n n n ^ 



2r 



Re 



E "(35,c;-dX) 



Re , N Ke '^^'o 



(2.29) 



'=;*^<^;-^'-'.)*^('-^K';^^ -Ahv^ -^)c% 



y ^ [mB rC' , + C- ,) - i^A rg^' , + ®' ,)] 



/ ■ ^ L m^ m+ 1 m~ 1 -^ m"- m+l m-l-'J 



561 



