Desai and Lieber 



PART 2 



CONSTRUCTION OF ANALYTICAL REPRESENTATION 

 OF VISCOUS FLOWS AROUND A CIRCULAR CYLINDER 



This part of the paper presents in outline some significant steps developed 

 in the application of the theory of Part 1 to the construction of an analytical 

 representation of viscous flows around a circular cylinder, based upon the com- 

 plete Navier-Stokes equations and realistic boundary conditions. 



Here, the subsidiary equations governing the coefficients A^Cr.t), B^(r,t), 

 C^(r,t), and Dn(r.t) of the stream functions i/y^ and ip^ as given by Eqs. (1.60) 

 and (1.61) respectively are obtained from the sets of equations for the first and 

 second iterations. Appropriate conditions at the two boundaries are respectively 

 obtained for these functions and their derivatives, with respect to r, from the 

 conditions of Eqs. (1.77) to (1.80), inclusive, and Eqs. (1.83) to (1.86), inclusive. 



FIRST ITERATION 



To obtain u^, Vj, and Pi which satisfy Eqs. (1.47), (1.48), and (1.49), and 

 the boundary conditions of Eqs. (1.77), (1.78), (1.83), and (1.84), we solve Eqs. 

 (1.50), together with the same boundary conditions, so as to obtain first \p^ and 

 hence ui and Vj. Then, from Eqs. (1.47) and (1.48), by integration we will ob- 

 tain pj. Using Eqs. (1.18) and (1.19) in Eq. (1.50), we get 



Re / 1, 



— (1 + — ) sin 



BV20^ r / 1 



Re 1 1 ) cos (9 



-— i = Re -— i . (2.1) 



Re 



Br Bt 



Now, in Eq. (1.60) we have 



CD 



0j = ^Ao(r,t) + Y, An(r.t) cosn^ + B„(r,t) sinn^ , 



^ n=l 



where A^, A^, and B^, are functions of r and t. These are the functions we 

 wish to determine. 



Using Eq. (1.60) and putting 



a^{v,t) - (Jo(r,t) + ^a;(r,t) 



a^(r,t) - a;(r,t) + ^a;(r,t) - ^(5„(r,t) (2.2) 



556 



