Studies on the Motion of Viscous Flows—Ill 



Bu. 



B2u, 



■2 sin 



-2 cos 



3u , 

 3r 



Bv 



2 cos 



3^2 3r W 



which should replace the values given in Eqs. (1.68). We also have 



1=0, 1^=0, (1-73) 



3v, 



?V. 



= , = +2 sin 



d0 



~ 2 cos 



(1.74) 



Using Eqs. (1.72), and Eqs. (1.73) with the rest of (1.68), and Eq. (1.74), we get 

 for r - 1 the following equations from Eqs. (1.47), (1.48), (1.51), (1.52), (1.55), 

 and (1.56): 



-4 cos^^ 



(-2 cos 6) 



'dv. 



+ 4 sins' cos 



( + 2 cos 5) 



3r 



6 COS 

 Re 



32 V, B V, 



Re\ Br2 Br 



r= 1 



^2 ;^ 



Br 



Re \ Br 



(1.75a) 



(1.75b) 



(1.75c) 



4 sing 

 Re 



(1.76a) 



(1.76b) 



Re 



■— ' +-^l . (1.76c) 



The term (+8 costs') of Eq. (1.75a) is the contribution of the linear convec- 

 tive terms in u^, v^ and their derivatives of the pressure gradient Bpj/Br at 

 r = 1. The term (-4 cos'^6) of Eq. (1.75b) is the contribution of the nonlinear 

 convective terms in Uj, Vj and their derivatives to the pressure gradient 

 Bp2/Br at r = 1. They are of the same order of magnitude, but have opposite 

 signs. Similarly, the left-hand side of Eq. (1.76a) is the contribution of the 

 linear convective terms in Uj, Vj and their derivatives to the pressure gradient 

 Bpj/Bg at r = 1, and the left-hand side of Eq. (1.76b) is the contribution of the 

 nonlinear convective terms in u^, Vj and their derivatives to the pressure gra- 

 dient Bp/Be'at r = 1. Both of these terms involve 



549 



