Lieber and Yajnik 

 Let H^ = P^ + i/2(ujUj). Then 



e^.^u-wj^ = H^i ; (16) 



It follows that ' r-'-- ' ■ r-:^ 



H^jUi = H^iWj = . " (17) 



That is, the symmetrical part of the total head (H^) does not vary along a stream- 

 line or a vorticity line. H^ is as a result constant in the surface of the stream- 

 line and the vorticity line. Further, from Eq, (16), we have 



e- (e -.uw. ) _ = e- H^„„ = 



imnv njk j k '' , — — 



, m 1 mn , nm 



That is, 



Since Uj i = wj . = 0, 



w„u „ „ - u w„ = . (18) 



This means that the increase of the velocity vector u,, along a vorticity 

 line is equal to the increase of the vorticity vector w^ along a streamline. 



Examples of Antisymmetric Flow 



The result wherein each component of vorticity obeys a heat conduction 

 equation is severe. This leads us to believe that the hypothesis of antisym- 

 metry is severe. A question naturally arises whether there are any flows 

 which satisfy the hypothesis. The following two examples show that there are 

 indeed flows where the conditions given by Eqs. (2) are met. 



Rectilinear Flows — In rectilinear flows, all particles move in parallel 

 straight lines. Let the common direction of motion be chosen as the X2 axis. 

 Then 



U2 = U2(Xj,X2,X3,t) , Uj = U3 = . 



Continuity requires that 



Thus, U2 is independent of Xj. Consequently, 



U2(Xj,X2,X3, t) = U2(Xj, -X2,X3,t) . 



Since Uj and U3 are identically zero, Eqs. (2) are satisfied. 



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