Lieber and Yajnik 



-P^ J + yV^Uj = -A^Xj ; 



"2,t + ^1^2, 1 + "2^2,2 + "3^2,3 = '^^2^^^ ' 



_ . "P, 2 + ^V^Uj = -A^x^ ; 



; 



-P , + vV''u. = = ; 



R2 R2 



,3 3 



and 



+ ^2,2 + "3,3 = 



Hence the above flow is dynamically possible. If we have a long pipe with rea- 

 sonably smooth entrance conditions, the actual flow would approximate the 

 above solution for low Reynolds numbers. 



We have additional information. From Thom's work, we know that vortices 

 behind a cylinder have approximately elliptical streamlines, and consequently 

 stream function is approximately symmetrical about one diameter [10]. K this 

 diameter is chosen as the xj axis, 



0(Xj,X2,X3, t) = i/;(Xj, - X2,X3, t) . 



Since u^ = -02 ^^^ "2 - '^.u "1 ^s odd and u^ is even in Xj. Equation (2) is 

 satisfied approximately. We can thus expect that the vorticity satisfies the heat 

 conduction equation. This was observed by Thom [lO]. Thus we have corre- 

 lated two observed features of a separated flow. 



Integration of Equations of Motion for Antisymmetrical Flow 



Having convinced ourselves about the physical significance of the family of 

 antisymmetrical flows, let us proceed to the task of integration of the equations 

 of motion. From Eqs. (12) and (14), 



"i.it + P'ii - ^v^Uii = p-ii = . (19) 



Hence the antisymmetric part of the pressure is harmonic. Let 



0= -/p- dt . (20) 



Then 



^.it = -p^ • (21) 



688 



