Studies on the Motion of Viscous Flows— I 



cannot increase indefinitely but must approach a finite value, say Sq . Applying 

 the mean value theorem to each component of 



t ds 



' ^0 , . . 



separately, and using Eq. (19), we get " ..-.V .' ^ ' [ 



I (x - (s - Sq ) a| < 3 e ( Sq - s) for Sq > s > Sj 



(20) 



smce 



ut = u, " (21) 



The expansion of velocity into a Taylor series and the use of Eq. (21) leads to 



or 



where 



lim |a"(Vu)|t = lim a'Vu 



a • (V u) = >v a. 



k = |a • V ul 



Hence we see by using Eq. (5) that the angular velocity of a line element in the 

 limiting direction a is zero. Clearly, the above argument, with minor modifi- 

 cations, applies when s decreases on approaching the point of zero velocity. 



Thus a streamline approaches the point of zero velocity in such a way that 

 if its unit tangent vector approaches a limiting value a , the angular velocity of 

 a line element along a is zero. This theorem shows that the question of whether 

 or not the streamline whirls around a point of zero velocity is directly related 

 to the angular velocity of line elements. 



The above results were obtained for use in subsequent sections of this 

 paper. It should be pointed out, however, that since these results throw light on 

 the connections between the rate of rotation of line and surface elements, vor- 

 ticity, curvature, and torsion of streamlines in a general way, they have a 

 broader significance than many other theorems in this paper. 



The dynamical significance of the angular velocities can be gauged from the 

 observation that the dissipation in a Newtonian fluid of constant properties can- 

 not be expressed in terms of invariants of vorticity, but is proportional to the 

 following invariant of rotation tensor: 



(4R. ., R. . ., - 3R. . . . R. .i,k)/6, 



*• i:jk i:jk i^JJ i:kk' 



as can be verified by using Eq, (7). 



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