Studies on the Motion of Viscous Flows — V 



Also 



^,iij = -i'PNii dt = ' (22) 



as P^ is harmonic. Substituting Eq. (21) in Eq. (12), we obtain 



or 



On account of Eq. (22), we have .-; ; ' 



("i-^,i).t = ^-v2(Ui-0 j) . (23) 



In other words, the flow can be decomposed into two parts. One part arises from 

 a potential ^. The other is such that each component obeys the heat conduction 

 equation. We then conclude that any antisymmetric flow obeying the Navier- 

 Stokes equations can be written as 



u. = i + u! (24) 



where 



0,ii - , , (25) 



and 



u'. = vV^u'. , (26) 



and 



u! i = . . .. . (27) 



Equation (18) imposes an additional condition on ^ and u'. Let 



Then 



w- = w| + e- 



(28) 



(29) 



ijk^.kj - "i • 

 Substituting Eq. (28) in Eq. (18), we obtain 



v/'d' +u' ) - (4> +u' ) w' = . (30) 



So the integration of the equations of motion amounts to finding a velocity field 

 u; [1] and a potential 4" satisfying the following conditions: 



.. = (25) 



689 



