Studies on the Motion of Viscous Flows— III 



The corresponding stream function is given by 



0o(^.^) = 



1 - 



(B) 



We can see from these expressions that as e^^ + e^ — * 0, e^ - e^ ^ 0, and h -^ 

 infinitely large distance, ^^ reduces to the previous solution. Even though we 

 happen to have a relation between h, e^, and e^, h cannot be calculated be- 

 cause we do not know e^ and e^. In general, we do not have such an explicit 

 relation. To see how c. and c, depend on h we may rewrite them as follows: 



Since the actual nature of e^j and e^ is not known, we may assume that the 

 ratio 6^ + €^/e^ - £y will generally be finite and that e^ + e^ is definitely 

 smaller than 2u„, in view of the fact that they each are smaller than the error 



of the instruments, 

 cally the values - G 



Hence c^ and C2 as functions of h approach asymptoti- 

 , and +u^a2 respectively as h becomes increasingly large. 



The pressure field is given by pQ(r,6') such that 



PnC'^'^) = 7 P^a 



2 cos 261 



-.2 \ 7.2 



2u„ 



+ P<x> + e. 



(C) 



This also reduces to the pressure field as obtained before when e^, = e^ - l^ - 0. 



The stream function in Eq. (B) and the pressure field in Eq. (C) adequately 

 represent the potential flow, though the conditions were applied at a finite dis- 

 tance h. 



The situation can now be generalized. We apply the conditions of uniform 

 flow to the general solution of the equations of motion at a finite distance^ say h. 

 We then obtain the constants of the solution as a function of the distance h. If 

 we find analytically or numerically that these constants are approaching limit- 

 ing values as the distance at which the conditions of uniform flow are applied is 

 increased, then we may take for solution that distance at which they approach 

 the limiting values, and consequently the final solution does not change signif- 

 icantly as the Physically Infinite Distance for the problem. Since the solution 

 does not change, the physical quantities like drag and pressure should also ex- 

 hibit asymptotic behavior with increasing h. 



From the above, we can state the condition of uniform flow as follows: 



515 



