studies on the Motion of Viscous Flows — III 



BVV, 



1 _d_ 

 ^ '66 



^'(^1+^2) 



B0, 



■" 3. 



V'C^ + '/'a) 



^3 



Fi7 (^o + ^l+'^2) 



1 S^V, 



1 B 



(00 + ^1 + '/'2) 



BV20. 



Re 



vv. 



1 



+ I — 



3s^2 =^^^0' 



r 30 Br 



1 302 ^^^2 



(1.58) 



We observe that u, v, and p are physical quantities. They are the field 

 variables in which we are interested. At any generic point P in the flow field 

 these quantities should have a definite set of values at a given time. Since the 

 choice of the zero direction for the polar axis from which 6 is measured is ar- 

 bitrary, and since after a complete rotation of the radius vector through an 

 angle of l-n radians we arrive at the same geometrical point from which we 

 started, any increase in 9 by multiples of 2v should not affect the values of u, 

 V, and p at any generic point in the field of any given time. This means that u, 

 V, and p should be periodic functions of 6 with a period of 2t7 radians. This 

 periodicity condition is expressed as follows: 



u (r,0, t) = u(r,e + 2n7r, t) 



v(r,5,t) = V (r,0+ 2n77, t) n = ±1, 2, ... 



p(r,6i, t) = p( r,0 + 2n7r, t) 



(1.59) 



Consequently, we can assume that the stream function i/'C r, 9, t) and hence 

 the functions u^, v^, p^, and /v, are also periodic functions of 9. Let us 

 therefore assume the following Fourier representations for 4i^, 4j^, and 1//3 re- 

 spectively: 



A ( r t ") '^ 



i/;j(r,0,t) = — '■ — + 2] A^C^'t) cosn0 + B^(r,t) sinn( 



2 „ - 1 



(1.60) 



0,(r,e,t) 



Co(r,t) 



+ Y\ C^(r , t) cos n(9 + D^(r, t) sinn(! 



(1.61) 



0,(r,0,t) = 



Eo(r,t) 



-(- 2] EnC'"'^) cosn0 + F^(r,t) sinni! 



(1.62) 



where A, B, C, D, E, and F stand for the coefficients of the Fourier repre- 

 sentations. 



We now impose suitable boundary conditions on the solutions of various 

 iterations. Let the first iteration velocity components Uj and Vj be such that 

 the sums (uq + Uj) and (v^ + Vj ) satisfy all the velocity conditions on the 



545 



