Studies on the Motion of Viscous Flows — III 



principle, these conditions together with an initial condition permit an examina- 

 tion of the transient flow. 



The choice of an initial condition is very important from the point of view 

 of the progressive evolution of the flow structure. A critical examination of the 

 governing questions reveals under what circumstances a flow can become time- 

 dependent. This being the case, a condition which is mathematically simple and 

 yet physically representative should be selected with care. In this work, the 

 equations of motion for a time-dependent motion are carefully examined without 

 attempting to solve them. Consequently, we have not formulated realistic initial 

 conditions. The time-independent case for which the boundary conditions of Eqs, 

 (1.16) suffice, is treated completely. •, .(■- . -,.1 ■ • 



POTENTIAL FLOW SOLUTION 



Let us consider the stream function 



0„ = 0o(r,0) . -(r -i) sin0 , (1.17) 



which leads to the velocity components 



u„= u,(r,^) = .i'^= -(l- -l)cos0 , ,.(1.18) 



V, = v„(r,9) = - — ° = + (l + -^) sine , (1.19) 



and the pressure field 



■ Po = Po(^.^)= 71 (2cos2e - -^) ■;.; (1.20) 



These functions in Eqs. (1.17), (1.18), (1.19), and (1.20) satisfy the following 

 equations: 



^"0 ^ ^!^ ^0^ 1 ^PQ 



3t ^ "° 3r ^ r 3^ ~ r ~ 2 Br 



dt ° 3r r 3^ r 2r d9 



9r ^ I" ^ •■ Be 



(1.21) 



and they are such that 



519 



